Methods and systems for computing trading strategies for use in portfolio management and computing associated probability distributions for use in option pricing

ABSTRACT

Exemplary methods and systems for creating uncorrelated trading strategies and deriving associated implied probability distributions of the price of an underlying financial instrument at future times are disclosed, applicable to stock market prices, interest rates, currency exchange rates, commodity prices and credit spreads.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.61/296,384, filed on Jan. 19, 2010, and entitled “METHOD AND SYSTEM FORCREATING TRADING STRATEGIES FOR USE IN PORTFOLIO MANAGEMENT AND DERIVINGASSOCIATED PROBABILITY DISTRIBUTIONS FOR USE IN OPTION PRICING,” whichis hereby incorporated by reference herein.

TECHNICAL FIELD

The disclosed technology is related to financial instruments. Particularembodiments, for example, include methods and systems for creatinguncorrelated trading strategies in a financial instrument and/or forcomputing a probability distribution of the future price of thefinancial instrument.

BACKGROUND

Financial market traders and investors at commercial and investmentbanks, hedge funds, other managed funds, pension funds, corporatetreasuries, insurance companies and elsewhere (including individualinvestors) have an interest in the use of options written on anunderlying financial instrument for investment trading and hedgingpurposes. For example, a company might wish to hedge against futureprice movements in oil or a currency exchange rate, and an investmentbank proprietary trader might wish to take a view (an investmentposition) on the future price movements of a stock. Accordingly, theaccurate valuation of options is important to a wide range of financialmarket participants. Further, option prices convey information aboutexpected future prices (e.g., future interest rates) that are ofinterest to other market observers, such as economic forecasters andcentral banks.

SUMMARY

Disclosed below are representative embodiments of methods, apparatus,and systems for computing uncorrelated trading strategies in a financialinstrument and/or for deriving a probability distribution of the futureprice of the financial instrument.

Certain embodiments of the disclosed technology provide a completeframework for incomplete markets. For example, some embodiments of thedisclosed technology produce complete option pricing in incompletemarkets, in that they explicitly model the nature of any incompletemarket, so as to determine the range of all possible prices for anoption on an underlying financial instrument (e.g., on a single asset, asingle asset class, or a fixed-weight portfolio).

Some embodiments of the disclosed technology use historical price data,not assumed forms for functions. For example, certain embodiments of thedisclosed technology do not assume specific functional forms (such asstochastic volatility or jumps) for asset price processes, probabilitydistributions, or risk premia. Rather, embodiments of the disclosedtechnology derive risk factors and associated risk premia from thehistorical prices for the underlying instrument through an expectationsgenerator for those prices (a generator of the present value pricingoperator).

Particular embodiments of the disclosed technology comprise anexpectations generator that generates expectations based on historicalprice data. For example, some embodiments of the disclosed technologyuse an expectations generator derived from a probability distribution ofshort-run future prices of the underlying instrument. The short-rundistribution can be based purely on the historical price data for theunderlying instrument, and, in one particular embodiment, is determinedby the technique of orthogonal series (independent components), namelyinitial estimates of risk factors.

Certain embodiments of the disclosed technology use risk factors derivedas functions of the underlying instrument price, which are thereforeusable as trading strategies in that instrument. For example,embodiments of the disclosed technology generate risk factors (finalestimates) that are explicit functions of the underlying instrumentprice, and hence so also are the associated hedging coefficients.Consequently, the risk factors can be used as trading strategies in theunderlying instrument, together with the hedging coefficients, so as toearn the risk premia (the expected excess returns) in a tradingportfolio (or pay it in an options hedging portfolio).

Some embodiments of the disclosed technology combine risk factors andrisk premia (the expected excess returns) to generate a futuremartingale probability distribution. For example, based purely on thehistorical price data for the underlying instrument, the risk factorsand risk premia can be combined in embodiments of the disclosedtechnology to generate a risk neutral (or “martingale”) probabilitydistribution of the underlying instrument price at a future time,corresponding to an option expiration date. This distribution can serveas an intermediate step to option pricing through the values of itsmoments.

Particular embodiments of the disclosed technology determine all futuremartingale distributions by a single function of a special mathematicalform. For example, consistent with the basis of historical price data,embodiments of the disclosed technology determine all possiblemartingale distributions at the future time through a single parameterfunction. In particular embodiments, the parameter function belongs to acertain mathematical class of functions called the “compactifiedNevanlinna class” and has two extremal values that determine upper andlower bounds to the price of any option on the underlying instrument.

Certain embodiments of the disclosed technology use the single functionto price new basket options. For example, the price of a basket(fixed-weight portfolio) option can be determined by embodiments of thedisclosed technology even when no options on that particular basket havebeen previously traded by assigning a value to the single parameterfunction derived from prices of options on the basket components.

Some embodiments of the disclosed technology determine martingaledistributions by a single scalar parameter. On the basis of both thehistorical underlying instrument prices and the contemporaneous optionprices, embodiments of the disclosed technology determine all possiblemartingale distributions at the future time through a single scalarparameter (a real number, possibly infinite). The scalar parameter hastwo extremal values that determine conditional upper and lower bounds tothe price of any option on the underlying instrument, conditional on thecontemporaneous option prices.

Particular embodiments of the disclosed technology determine the optimaldistribution (at each future time) for pricing any option by optimizingthe scalar parameter to option prices. For example, embodiments of thedisclosed technology determine an optimal martingale distribution (onethat implies option prices that best or nearly best approximate thecontemporaneous option prices) by optimizing a single scalar parameter.

Certain embodiments of the disclosed technology use algebraicalgorithms, not difficult-to-solve differential or stochastic equations.For example, embodiments of the disclosed technology employ onlyalgebraic algorithms and do not, therefore, bear the computationalburden incurred in solving partial differential equations, partialintegro-differential equations, or their counterparts in stochasticdifferential equations.

Some embodiments of the disclosed technology are configured to displaygraphically (e.g., on a display device) the trading strategies (e.g.,the values of the positions in the financial instrument as a function ofits price) and the associated hedging coefficients and excess returnsand, at each selected future time, the optimal probability distribution,and the extremal and conditional extremal probability distributions.

In certain embodiments, uncorrelated trading strategies are created andassociated martingale probability distributions are computed by defininga parameterized probability distribution model and then calculatingorthogonal polynomials in accordance with the historical price data forthe underlying instrument, calculating a transformation matrix inaccordance with the underlying instrument data, and determining thevalues of the distribution parameters. The values of the distributionparameters can be determined in accordance with, in initial estimate,the transformation matrix, and the contemporaneous price data for theplurality of options and, in subsequent iterations, the transformationmatrix, the options data, and an auxiliary transformation matrix itselfdetermined in accordance with the options data in conjunction with theunderlying instrument data. All computations can be carried out by meansof algebraic algorithms and, therefore, there is no need to solve anypartial differential equations or partial integro-differentialequations. Furthermore, the trading strategies (optionally together withhedging coefficients and excess returns) and/or the martingaleprobability distributions (optionally together with upper and lowerbounds) can further be graphically displayed.

One of the disclosed embodiments is a computer-implemented method thatcomprises receiving historical price data, the historical price dataindicating one or more historical prices for a financial instrument;receiving maturity data, the maturity data indicating one or moreexpiration dates, each of the one or more expiration dates being anexpiration date for one or more options on the financial instrument;selecting an expiration date from the one or more expiration dates;computing a set of two extremal martingale probability distributionsbased at least in part on the historical price data and the selectedexpiration date, each of the two extremal martingale probabilitydistributions indicating probabilities of possible prices for thefinancial instrument at the selected expiration date, wherein each ofthe two extremal martingale probability distributions is determined by afractional linear transformation of a distribution parameter function;and storing the set of two extremal martingale probabilitydistributions. In certain implementations, the act of computing the setof two extremal martingale probability distributions comprises settingthe value of the distribution parameter function to each of twopre-determined fixed scalar values. In some implementations, thecomputing the set of two extremal martingale probability distributionscomprises computing a representative martingale probability distributionusing the historical price data and the selected expiration date, therepresentative martingale probability distribution indicatingprobabilities of possible prices for the financial instrument at theselected expiration date; computing one or more mathematical moments ofthe representative martingale probability distribution; computing thefractional linear transformation of the distribution parameter function,the fractional linear transformation being based at least in part on theone or more mathematical moments; assigning values to the distributionparameter function; and computing the set of two extremal martingaleprobability distributions based at least in part on the fractionallinear transformation and the values assigned to the distributionparameter function. The mathematical moments of the representativemartingale probability distribution can be computed by an Esschertransform of an empirical probability distribution. Further, the set oftwo extremal martingale probability distributions can be based on theone or more mathematical moments of the representative martingaleprobability distribution, and not on any other use of the historicalprice data. In certain implementations, the act of computing the set oftwo extremal martingale probability distributions is performed usingonly algebraic manipulations. In some implementations, the computing theset of two extremal martingale probability distributions is performedwithout using either stochastic computations or differential equationcomputations. In some implementations, the method further comprisesreceiving a forward price or discount factor for the financialinstrument at the selected expiration date, and the computing the set oftwo extremal martingale probability distributions is further based atleast in part on the forward price or the discount factor. In certainimplementations, the set of two extremal martingale probabilitydistributions comprises a first martingale probability distributionindicating lower bounds of possible prices for an option on thefinancial instrument at the selected expiration date, the lower boundsbeing independent of current option prices for the financial instrument;and a second martingale probability distribution indicating upper boundsof possible prices for an option on the financial instrument at theselected expiration date, the upper bounds being independent of currentoption prices for the financial instrument.

Another one of the disclosed embodiments is a computer-implementedmethod that comprises receiving historical price data, the historicalprice data indicating one or more historical prices for a financialinstrument; receiving maturity data, the maturity data indicating one ormore expiration dates, each of the one or more expiration dates being anexpiration date for one or more options on the financial instrument;selecting an expiration date from the one or more expiration dates;receiving option prices, the option prices indicating a respectivemarket price for each of one or more available options on the financialinstrument, each of the option prices being contemporaneous orsubstantially contemporaneous with one another, and each of theavailable options having an expiration date coinciding with the selectedexpiration date; computing a set of two conditional extremal martingaleprobability distributions based at least in part on the historical pricedata, the selected expiration date, and the option prices, each of thetwo conditional extremal martingale probability distributions indicatingconditional probabilities of possible prices for the financialinstrument at the selected expiration date, the conditionalprobabilities being conditional on the option prices, wherein each ofthe two conditional extremal martingale probability distributions isdetermined by a fractional linear transformation of a distributionparameter function; and storing the set of two conditional extremalmartingale probability distributions. In certain implementations, theact of computing the set of two conditional extremal martingaleprobability distributions comprises computing initial estimates ofapproximating martingale probability distributions based on test scalarvalues of the distribution parameter function; and setting the value ofan interpolation parameter function to each of two pre-determined fixedscalar values. In some implementations, the act of computing the set oftwo conditional extremal martingale probability distributions comprisescomputing a representative martingale probability distribution using thehistorical price data, the representative martingale probabilitydistribution indicating probabilities of possible prices for thefinancial instrument at the selected expiration date; computing one ormore mathematical moments of the representative martingale probabilitydistribution; computing the fractional linear transformation of thedistribution parameter function, the fractional linear transformationbeing based at least in part on the one or more mathematical moments;assigning values to the distribution parameter function; assigningvalues to an interpolation parameter function; and computing the set oftwo conditional extremal martingale probability distributions based atleast in part on the fractional linear transformation, the valuesassigned to the distribution parameter function, and the values assignedto the interpolation parameter function. The one or more mathematicalmoments of the representative martingale probability distribution can becomputed by an Esscher transform of an empirical probabilitydistribution. Further, the set of two conditional extremal martingaleprobability distributions can be based on the one or more mathematicalmoments of the representative martingale probability distribution, andnot on any other use of the historical price data. In someimplementations, the computing the set of two conditional extremalmartingale probability distributions is performed using only algebraicmanipulations. In certain implementations, the computing the set of twoconditional extremal martingale probability distributions is performedwithout using either stochastic computations or differential equationcomputations. In some implementations, the method further comprisesreceiving financial instrument price data indicating one or more marketprices for the financial instrument, each of the one or more marketprices being pairwise contemporaneous or substantially contemporaneouswith corresponding ones of the option prices, and the act of computingthe set of two conditional extremal martingale probability distributionsis further based at least in part on the financial instrument pricedata. In certain implementations, the method further comprises receivinga forward price or discount factor for the financial instrument at theselected expiration date, and the act of computing the set of twoconditional extremal martingale probability distributions is furtherbased at least in part on the forward price or the discount factor. Insome implementations, the set of two conditional extremal martingaleprobability distributions comprises a first martingale probabilitydistribution indicating lower bounds of possible prices for an option onthe financial instrument at the selected expiration date, the lowerbounds being dependent on the option prices for the financialinstrument; and a second martingale probability distribution indicatingupper bounds of possible prices for an option on the financialinstrument at the selected expiration date, the upper bounds beingdependent on the option prices for the financial instrument.

Another one of the disclosed embodiments is a computer-implementedmethod that comprises receiving historical price data, the historicalprice data indicating one or more historical prices for a financialinstrument; receiving maturity data, the maturity data indicating one ormore expiration dates, each of the one or more expiration dates beingthe expiration date for one or more options on the financial instrument;selecting an expiration date from the one or more expiration dates;receiving option prices, the option prices indicating a market price foreach of one or more available options on the financial instrument, eachof the option prices being contemporaneous or substantiallycontemporaneous with one another, and each of the available optionshaving an expiration date coinciding with the selected expiration date;computing an approximating martingale probability distribution based atleast in part on the historical price data, the selected expirationdate, and the option prices, the approximating martingale probabilitydistribution indicating probabilities of possible prices for thefinancial instrument at the selected expiration date, the approximatingmartingale probability distribution being indicative of option pricesthat approximate the option prices, wherein the approximating martingaleprobability distribution is determined by a fractional lineartransformation of a distribution parameter function; and storing theapproximating martingale probability distribution. In someimplementations, the method further comprises computing estimated optionprices from the approximating martingale probability distribution. Incertain implementations, the act of computing the approximatingmartingale probability distribution comprises computing initialestimates of the approximating martingale probability distribution basedon test scalar values of the distribution parameter function; andcomputing the approximating martingale probability distribution based ontest scalar values of an interpolation parameter function. In someimplementations, the act of computing the approximating martingaleprobability distribution comprises computing a representative martingaleprobability distribution using the historical price data, therepresentative martingale probability distribution indicatingprobabilities of possible prices for the financial instrument at theselected expiration date; computing one or more mathematical moments ofthe representative martingale probability distribution; computing thefractional linear transformation of the distribution parameter function,the fractional linear transformation being based at least in part on theone or more mathematical moments; assigning values to the distributionparameter function; assigning values to the interpolation parameterfunction; and computing the approximating martingale probabilitydistribution based at least in part on the fractional lineartransformation, the values assigned to the distribution parameterfunction, and the values assigned to the interpolation parameterfunction. The one or more mathematical moments of the representativemartingale probability distribution can be computed using an Esschertransform of an empirical probability distribution. Further, theapproximating martingale probability distribution can be based on theone or more mathematical moments of the representative martingaleprobability distribution, and not on any other use of the historicalprice data. In certain implementations, the act of computing theapproximating martingale probability distribution is performed usingonly algebraic manipulations. In some implementations, the act ofcomputing the approximating martingale probability distribution isperformed without using either stochastic computations or differentialequation computations. In certain implementations, the method furthercomprises receiving financial instrument price data indicating one ormore market prices for the financial instrument, each of the one or moremarket prices being pairwise contemporaneous or substantiallycontemporaneous with corresponding ones of the option prices, and thecomputing the approximating martingale probability distribution isfurther based at least in part on the financial instrument price data.In some implementations, the method further comprises receiving aforward price or discount factor for the financial instrument at theselected expiration date, and the computing the approximating martingaleprobability distribution is further based at least in part on theforward price or the discount factor.

Another one of the disclosed embodiments is a computer-implementedmethod that comprises receiving historical price data, the historicalprice data indicating one or more historical prices for a financialinstrument; receiving a time horizon, the time horizon being not lessthan a shortest period between successive prices in the historical pricedata; computing one or more trading strategies based at least in part onthe historical price data and the time horizon, each of the one or moretrading strategies comprising a function that relates a value of therespective trading strategy to a price of the financial instrument,wherein each of the one or more trading strategies is determined by asecond order finite difference. In some implementations, the methodfurther comprises, for each of the one or more trading strategies,computing an excess return, one or more hedging coefficients, or both anexcess return and one or more hedging coefficients. In certainimplementations, the act of computing the one or more trading strategiescomprises computing a time horizon probability distribution indicatingprobabilities of possible prices for the financial instrument at thetime horizon; computing one or more mathematical moments of the timehorizon probability distribution; and computing one or more coefficientsof the second order finite difference, the one or more coefficientsbeing based at least in part on the mathematical moments of the timehorizon probability distribution. The one or more mathematical momentscan be computed using an orthogonal series method. In someimplementations, the time horizon is a first time horizon, and the actof computing the one or more trading strategies comprises receiving asecond time horizon (the second time horizon being different than thefirst time horizon); receiving a forward price or discount factor forthe financial instrument at the received second time horizon; computinga representative martingale probability distribution using the forwardprice or the discount factor, the representative martingale probabilitydistribution indicating probabilities of possible prices for thefinancial instrument at the second time horizon; computing one or moremathematical moments of the representative martingale probabilitydistribution; and computing one or more coefficients of the second orderfinite difference, the one or more coefficients being based at least inpart on the mathematical moments of the representative martingaleprobability distribution. The one or more mathematical moments of therepresentative martingale probability distribution can be computed usingan Esscher transform of an empirical probability distribution. Further,the one or more trading strategies can be computed as orthogonalfunctions. In certain implementations, the method further comprisescomputing hedging coefficients or excess returns associated with one ormore of the trading strategies as explicit functions of a price of thefinancial instrument. In some implementations, the trading strategiesare computed without using option price data. In certainimplementations, the computing the one or more trading strategies isbased on one or more mathematical moments of the time horizonprobability distribution, and not on any other use of the historicalprice data. In some implementations, the computing the one or moretrading strategies is based on one or more mathematical moments ofrepresentative martingale probability distribution, and not on any otheruse of the historical price data. In certain implementations, thecomputing the one or more trading strategies is performed using onlyalgebraic manipulations. In some implementations, the computing the oneor more trading strategies is performed without using stochasticcomputations or differential equation computations. In certainimplementations, the time horizon probability distribution is based atleast in part on the historical price data.

Another one of the disclosed embodiments is a computer-implementedmethod that comprises receiving historical price data, the historicalprice data indicating one or more historical prices for a financialinstrument; receiving maturity data, the maturity data indicating amaturity date for one or more options on the financial instrument;receiving contemporaneous price data, the contemporaneous price dataindicating a contemporaneous or substantially contemporaneous marketprice for each of one or more available options on the financialinstrument, each of the one or more available options having a maturitydate coinciding with the received maturity date; computing a set of twoor more extremal martingale probability distributions, a set of two ormore conditional extremal martingale probability distributions, and oneor more approximating martingale probability distributions based atleast in part on the historical price data, the maturity data, and thecontemporaneous price data, wherein each of the two extremal martingaleprobability distributions, each of the two conditional extremalmartingale probability distributions, and each of the one or moreapproximating martingale probability distributions is determined by afractional linear transformation of a distribution parameter function;and causing the set of two or more extremal martingale probabilitydistributions, the set of two or more conditional extremal martingaleprobability distributions, and the one or more approximating martingaleprobability distributions to be displayed on a display device. Incertain implementations, the method further comprises receiving a timehorizon, the time horizon being not less than a shortest period betweensuccessive prices in the historical price data; computing one or moretrading strategies based at least in part on the historical price dataand the time horizon, each of the one or more trading strategiescomprising a function that relates a value of the respective tradingstrategy to a price of the financial instrument; and causing the one ormore trading strategies to be displayed on the display device. Themethod can further comprise, for at least one of the one or more tradingstrategies, computing an excess return, one or more hedgingcoefficients, or both an excess return and one or more hedgingcoefficients; and causing the excess return, the one or more hedgingcoefficients, or both the excess return and the one or more hedgingcoefficients to be displayed on the display device. In someimplementations, an identification of the financial instrument and thematurity date are received from a user via a user interface. In certainimplementations, the method further comprises receiving a forward priceor discount factor for the financial instrument at the selectedexpiration date, and the computing the set of two or more extremalmartingale probability distributions, the set of two or more conditionalextremal martingale probability distributions, and the one or moreapproximating martingale probability distributions is further based atleast in part on the forward price or the discount factor.

Any of the disclosed embodiments can be implemented in a computer systemcomprising a processor, non-transitory data storage medium, and otherelements.

Non-transitory computer-readable media (e.g., one or more non-transitorycomputer-readable media) storing computer-executable instructions whichwhen executed by a computer cause the computer to perform any of thedisclosed methods or method acts, alone or in various combinations andsubcombinations with one another, are also disclosed herein andconsidered to be within the scope of this disclosure. Further,non-transitory computer-readable media (e.g., one or more non-transitorycomputer-readable media) storing any of intermediate or final resultsgenerated at least in part by performing any of the disclosed methods ormethod acts, alone or in various combinations and subcombinations withone another, are also disclosed herein and considered to be within thescope of this disclosure. Also disclosed herein and considered to bewithin the scope of this disclosure is a computer comprising a processorand memory, the computer being configured to perform any of thedisclosed methods or method acts, alone or in various combinations andsubcombinations with one another. Also disclosed herein and consideredto be within the scope of this disclosure is a display device displayingone or more intermediate or final results generated at least in part byperforming any of the disclosed methods or method acts, alone or invarious combinations and subcombinations with one another.

The foregoing and other objects, features, and advantages of thedisclosed technology will become more apparent from the followingdetailed description, which proceeds with reference to the accompanyingfigures.

BRIEF DESCRIPTION OF THE FIGURES

FIGS. 1A-1B show a flowchart of an exemplary technique for determining aconditional returns distribution by orthogonal series in accordance withan embodiment of the disclosed technology.

FIGS. 2A-2B show a flowchart of an exemplary method for creating atrading strategy, and associated hedging coefficients, in accordancewith an embodiment of the disclosed technology.

FIG. 3 shows a graph showing example trading strategies.

FIG. 4 shows a flowchart of an exemplary method for computing amartingale probability distribution and its optimal and conditionaloptimal upper and lower bounds in accordance with an embodiment of thedisclosed technology.

FIGS. 5A-5B show a flowchart of an exemplary technique for determining afinal estimate of a martingale probability distribution by constructinga second value of the Pick matrix in accordance with an embodiment ofthe disclosed technology.

FIGS. 6A-6B show a flowchart illustrating a generalized method forgenerating multiple martingale probability distributions for an optionor a group of options having a common maturity date.

FIG. 7 is a flowchart illustrating an exemplary method for generatingone or more of trading strategies, hedging coefficients, or expectedexcess returns for an underlying financial instrument.

FIG. 8 shows a schematic block diagram of a computing environment thatcan be used to implement embodiments of the disclosed technology.

FIG. 9 shows a schematic block diagram of a first network topology thatcan be used to implement embodiments of the disclosed technology.

FIG. 10 shows a schematic block diagram of a second network topologythat can be used to implement embodiments of the disclosed technology.

DETAILED DESCRIPTION I. General Considerations

Disclosed below are representative embodiments of methods, apparatus,and systems for computing trading strategies in a financial instrumentand/or for computing a probability distribution of the future price ofthe financial instrument. The disclosed methods, apparatus, and systemsshould not be construed as limiting in any way. Instead, the presentdisclosure is directed toward all novel and nonobvious features andaspects of the various disclosed embodiments, alone and in variouscombinations and subcombinations with one another. Furthermore, anyfeatures or aspects of the disclosed embodiments can be used in variouscombinations and subcombinations with one another. For example, one ormore method acts from one embodiment can be used with one or more methodacts from another embodiment and vice versa. The disclosed methods,apparatus, and systems are not limited to any specific aspect or featureor combination thereof, nor do the disclosed embodiments require thatany one or more specific advantages be present or problems be solved.

Although the operations of some of the disclosed methods are describedin a particular, sequential order for convenient presentation, it shouldbe understood that this manner of description encompasses rearrangement,unless a particular ordering is required by specific language set forthbelow. For example, operations described sequentially may in some casesbe rearranged or performed concurrently. Moreover, for the sake ofsimplicity, the attached figures may not show the various ways in whichthe disclosed methods can be used in conjunction with other methods.Additionally, the description sometimes uses terms like “determine” and“generate” to describe the disclosed methods. These terms are high-levelabstractions of the actual operations that are performed. The actualoperations that correspond to these terms may vary depending on theparticular implementation and are readily discernible by one of ordinaryskill in the art.

II. Introduction to the Disclosed Technology

In general, an option is a derivative whose contractual payoff isdefined in terms of the price of an underlying financial instrument,which may be a futures price or an index value, as well as the price ofan individual asset or basket of assets drawn from the equity, interestrate, currency, commodity, credit, or other financial markets. An optionon a specified underlying financial instrument is contractuallycharacterized by, inter alia, its maturity (or expiration) date T, itsstrike (or exercise) price K, and its type, such as European orAmerican. A European call option gives the holder the right to buy theunderlying financial instrument (in a pre-specified quantity) at thestrike price on the maturity date. A European put option similarly givesthe right to sell. For brevity, the present disclosure will refer simplyto calls and puts when the type of an option is European. It is to beunderstood, however, that the technology is not limited by such usageand can be applied to any type of option. Options of various types, themarkets in which they are traded, the underlying financial instrumentsand the pricing principles are described in J. C. Hull, Options,Futures, and Other Derivatives (2008). The disclosed technology can beadapted for use for any such type, market, or financial instrument.

One basic approach to pricing options is the Black-Scholes-Mertonframework. In this approach, it is assumed that the market satisfies noarbitrage. In other words, the market precludes the possibility of asure gain without risk of loss from trading strategies that employfinite (bounded) capital. It is postulated that the price process,written S_(t) where t denotes time, of the underlying financialinstrument follows a geometric Brownian motion, entailing a lognormaldistribution at any future time. This means that log S_(t), thelogarithm of S_(t), at a future time follows a normal (or Gaussian)distribution with a constant volatility (the square root of variance),denoted σ. On this basis, along with other assumptions, theBlack-Scholes formula for a stock option states that the price at time tof a call with strike K and maturity T denoted C_(t)(K,T), or simplyC(K,T) or C(K) or C, is a certain function of the underlying priceS_(t), the time to expiry (T−t), the strike price K, certain otherparameters (the cash market interest rate and the dividend rate), andthe volatility σ. Whatever the model, the values of the first and secondderivatives of option price C with respect to underlying instrumentprice S_(t) are important in order to hedge or replicate options, andare referred to as the delta and gamma coefficients, respectively.

Under the assumption of no arbitrage, a set of prices {C(K,T)} (where {} denotes a set) of various strike prices {K} and fixed maturity T arerelated to a probability distribution over the underlying price atmaturity date T. See, e.g., S. A. Ross, “Options and Efficiency” inQuarterly Journal of Economics, vol. 90(1), pages 75-89 (1976) and D. T.Breeden and R. H. Litzenberger, “Prices of State-Contingent ClaimsImplicit in Option Prices” in Journal of Business, vol. 51(4), pages621-651 (1978). The Breeden-Litzenberger prescription states therelation:

${\frac{\partial^{2}{C\left( {K,T} \right)}}{\partial K^{2}} = {Q\left( {d\; K} \right)}},$

that the second partial derivative of the call price with respect to thestrike price equals the measure (probability distribution) Q, pertainingto S_(T), evaluated at the infinitesimal increment dK. More generally,taking a triplet of equally spaced strike prices spanning a finiteincrement, the relation holds for the second difference in call pricesand the measure of the corresponding second difference in strike prices.

The measure Q (with fixed maturity T understood) is called themartingale measure, also referred to as the risk neutral distribution,because it pertains not to an actual observable distribution, but ratherto an adjusted distribution, with the adjustment corresponding to fairmarket (martingale) valuation or to risk aversion of a hypothetical“representative investor”. Methods to value options typically dependeither implicitly or explicitly on evaluating Q (as a distribution overthe price of the underlying financial instrument). According to modernfinance theory, if there are no arbitrage opportunities in a financialmarket, corresponding to any given buy-and-hold price distribution P,representing expectations of the price that will be observed at date T,there are one or more martingale measures Q, the set being denoted {Q}.The corresponding measures {Q} are equivalent to P, meaning they sharewith P the same null sets (the values of price for which the probabilitymeasure is zero).

Another theorem of asset pricing asserts that the completeness of thefinancial market is equivalent to the uniqueness of Q and, byimplication, if the market is incomplete then there exists {Q}, a set ofmore than one measure. By complete market is meant that the marketedfinancial instruments can be traded in strategies that employ finite(bounded) capital so as to replicate any bounded function defined on theprobability space characteristic of the market. Intuitively, a completemarket is one where all risks can be hedged by trading the existingfinancial instruments (it is possible to form a perfect hedge). In acomplete market, an option price C is uniquely determined by Q. In anincomplete market, a perfect hedge is no longer possible. Price C is nolonger uniquely determined, but rather falls within a range with upperand lower bounds determined by the set {Q}.

Through the Black-Scholes formula, a given value of volatilitydetermines a price C (taking other parameters to be fixed). Conversely,a given price C(K) determines a volatility σ(K), called the impliedvolatility (with fixed T understood). Hence, the Black-Scholes formulacan be viewed as a transformation of price into implied volatility, inthe same way that the yield-to-maturity (or internal rate-of-return)formula transforms a bond price into a yield. When implied volatilityσ(K) is plotted against K (as abscissa), in general it is not constant(in other words, it is not a horizontal straight line), contrary to whatwould be the case if the Black-Scholes formula were an exact descriptionof market behavior. The typically convex shape of the plot is referredto as the implied volatility “smile” or “smirk”. Further, σ(K) is notconstant through time t. A detailed empirical examination of impliedvolatility indicates that the form of Q is skewed by a fat left tail andhas excess kurtosis (an indicator of the fourth moment relative tovariance) in comparison with a lognormal distribution. Impliedvolatility exhibits volatility clustering through time (e.g., highlevels of volatility tend to occur together), since price increments areautocorrelated, especially in the short run, through squared logreturns. Further, implied volatility forms a well-defined volatilitysurface when plotted against K and T, but the surface changesperiodically in random fashion.

To model such behavior for the purpose of option pricing, a number ofdifferent approaches can be used. In one approach, a specific martingaleprice process other than geometric Brownian motion is postulated (orpossibly inferred through adjustment to a postulated observable priceprocess). Then the process is used to simulate values of the impliedvolatility at maturities { T}, and the associated volatility surface isfitted to market values by optimizing the parameters of the modelprocess. In a second approach, a specific functional form other thanlognormal is postulated for Q (again with maturity T understood). Then Qis used to price options (of the corresponding maturity T) and theresultant prices are fitted to market values by optimizing theparameters of the model distribution. In the first approach, there is animplied Q, and only one Q, generated by the martingale process andsatisfying the Breeden-Litzenberger prescription.

One example of the first approach is implied binomial and trinomial treemodels, which replace geometric Brownian motion by a more generaldiffusion process. A second example is the Merton jump-diffusion model,which postulates a combination of a diffusion process and a jump processcharacterized by three parameters: arrival rate (the number of jumps perunit time), jump volatility, and jump drift. A third example isstochastic volatility models, which postulate a stochastic process forvolatility, e.g. in the Hull and White model an independent geometricBrownian motion with constant coefficients. More complex processes forvolatility include regime switching and jump-diffusion.

One example of the second approach is the Jarrow-Rudd parametric model,which makes adjustments to the lognormal distribution so as to match thefirst four cumulants (in effect, moments) of Q, as inferred fromempirical data. A second example is the Jackwerth-Rubinstein smoothfitting model, which constrains the smoothness of Q according to acertain criterion, while maximizing a good fit to empirically givenoption prices.

Under the first approach, more generally, combinations of the varioustypes of processes are employed. The empirical performance of a varietyof models suggests that it is useful to allow certain variables in thestochastic price process themselves to be modeled as stochasticprocesses: jumps, volatility, and interest rates. Stochastic jumps andvolatility can be important to reproducing the smile, including thesmile on short-term options, while stochastic interest rates enhance thepricing fit of long-term options. In this way, the price process becomesdoubly stochastic with respect to these variables, which are “latent”(not directly observable). The wider significance of latent variables inoption pricing models, which is reinforced by other financial marketevidence, is that markets are incomplete. When markets are incomplete,there are risks that are not separately traded.

The condition of no arbitrage is typically imposed on the models in therequirement that Q is a martingale or equivalent martingale measure (avalid probability distribution with predetermined null sets), asdistinct from the joint requirement that Q is an equivalent martingalemeasure and is derivable from the given observable price distributiondata P. The former requirement imposes only a weak constraint on Q. Intechnical terms, this means that the set {Q} is derived not from P, butfrom any distribution of the same mean and null sets as P (from thatequivalence class of distributions). It is required that there exists aQ that is positive except on the null sets or, loosely speaking, thatoption price C(K,T) be non-decreasing in T and smile σ(K) be convex inK, at fixed T, with slope varying between −1 and zero.

A central issue in empirical tests of option pricing models isconsistency between the two data sets: observed time-series data of theunderlying asset price (possibly augmented by time series of optionprices) and cross-sectional option prices (at a point in time, optionprices grouped by common maturity), making due allowance for riskadjustment between the two. Two lines of inquiry ask “what does a modelestimated from time series data imply about option pricing?” and“conditional on a given cross-section of option prices, what does themodel predict about the joint behavior of asset and option prices (e.g.,joint returns)?”

An example responding to the second question is that option returns area levered function of asset returns, the function depending on optionprice. Experience with stochastic volatility models, however, shows thatderiving plausible model parameters from time series data gives a poorfit to smiles compared to unconstrained parameters. More generally, in avariety of models, combining cross-sectional options data withtime-series underlying asset price data effectively degenerates into atwo-stage procedure, with option prices implying Q and realizations oflatent variables, and time series asset prices implying the remainingparameters of the model.

These difficulties become acute in attempting to value risk premia forlatent variables, such as stochastic jumps, volatility, and interestrates, which are not directly priced by other traded assets. In someearly models, the risk premium was assumed to be zero (e.g., for jumprisk in the Merton jump-diffusion model, and for stochastic volatilityin the Hull and White model), by virtue of the assumption that theunhedged risk was diversifiable. In other models, the risk premium isassumed to take a specific parametric form and is fitted as a residual,e.g., in affine models the pricing kernel connecting P and Q is assumedto be exponential affine in the state vector. The common feature ofthese assumptions is that they impose pseudo-completeness on anincomplete market (e.g., unhedged risk becomes diversifiable or P isuniquely adjusted by the risk aversion of an implicit representativeinvestor).

One of the costs of pseudo-completeness is the absence of a rationalno-arbitrage basis for determining optimal upper and lower bounds to anoption price. To avoid such pseudo-completeness, some have evenabandoned the no-arbitrage paradigm and attempted to infer asset pricebounds from another criterion (e.g., “good deal” excess return-to-riskratios).

Numerical methods for implementing option models use, most often, one ofthree empirical techniques: Fourier inversion; finite differencemethods; or Monte Carlo simulation. Calibrating models to market data isa computationally demanding task. The reason is that one must computenumerical solutions to partial differential equations (or, with latticemethods, stochastic differential equations) and, in the case of priceprocesses with jumps, solutions to partial integro-differentialequations. The computational burden can be important because optionpricing models are typically updated daily or more frequently, and theresults are needed quickly during “real time” trading sessions. Aninstance of the consequent limitations is Monte Carlo simulation, whichoften is run to determine only the mean of an option price under thedistribution Q.

Associated to option pricing through each martingale measure Q (for agiven underlying financial instrument) is a set of trading strategies.These strategies are defined by the property that successive strategies,starting from the two strategies of invest in the cash asset andbuy-and-hold the underlying financial instrument, are uncorrelated underthe probability measure Q. The absence of correlation can be desirablebecause it enables portfolio managers to control risk and return moreaccurately. In particular, it means that a portfolio manager who iswilling to bear higher risk can aim to earn systematically higherreturns. In recent years (especially since about 2003 on), hedge fundshave become prominent active traders in a wide variety of financialmarkets, and many have earned superior returns against pre-specifiedbenchmarks (such as the S&P 500 index). Empirical investigation suggeststhat hedge funds, including the so-called commodity trading advisors(CTAs), employ just such uncorrelated trading strategies. Efforts toexplain and replicate hedge fund trading strategies have beenessentially empirical, in that they seek to reproduce (e.g., throughregression techniques) observed hedge fund returns on the basis ofselected asset class returns in “liquid” (easily traded) assets. Suchreplication is important because hedge funds charge high management andperformance fees, typically are illiquid (they restrict withdrawal ofinvestment funds), and are opaque in that they limit disclosure of theirproprietary trading strategies.

Further, the set of trading strategies provides an optimal hedge in anoptions portfolio, in the sense that a finite number of such strategies,if they were traded as derivative financial instruments in africtionless market, would be able to hedge any options exposures almostcompletely. The possibility of optimal hedging has not been generallyrecognized by financial market investors, however. Optimal hedging isdesirable because some banks, investment banks and other financialintermediaries hold large portfolios of options, acquired in the courseof their customer flow of trading business, which constitute substantialrisk exposures. The intermediaries seek to hedge these risk exposuresthrough continuous hedging, day-by-day. The trading strategies and theoption hedging strategies follow exactly the same trading pattern andmay be considered to be one and the same thing. In practice, they differin the way in which the time horizon for the strategy is determined. Thetrading strategy time horizon is based on a trading view as to whencertain events may occur. The option hedging strategy time horizon isbased on the maturities of the options to be hedged.

Accordingly, it is desirable to provide improved methods and systems forcreating uncorrelated trading strategies in the underlying financialinstrument, and for computing associated probability distributions todetermine option prices.

Disclosed herein are embodiments of methods, apparatus, and systems forcreating uncorrelated trading strategies in a financial instrument andcomputing probability distributions of the future price of the financialinstrument. The disclosed embodiments can be used to realize a number ofpossible advantages. For example, in certain embodiments, the disclosedmethods and systems do not, in effect, discard much of the informationcontained in historical time series data of the underlying asset price,but rather utilize that data in option pricing, without the need toselect a specific parametric functional form for either the martingalemeasure Q, or the price process, or the risk premium (risk aversion)factor. In some embodiments, the disclosed methods and systems do notignore or finesse the incomplete nature of the financial market, butrather incorporate it in a joint no-arbitrage requirement that Q is anequivalent martingale measure and is derivable from the data-determinedP, so as to provide rational optimal bounds to option prices. In certainembodiments, the disclosed methods and systems significantly reduce thecomputational burden of option pricing. In some embodiments, thedisclosed methods and systems generate explicit formulae (and relatedhedging parameters) for uncorrelated trading strategies in theunderlying financial asset (instrument), independently of whatever mightbe inferred from hedge fund trading strategies or financial intermediaryoption hedging strategies.

III. Exemplary Embodiments for Computing Trading Strategies for Use inPortfolio Management and/or for Computing Probability Distributions

Embodiments of the disclosed technology are capable of computing tradingstrategies for use in portfolio management (trading portfolios), and/orfor computing associated martingale probability distributions for use inpricing financial derivatives. Embodiments of the disclosed technologywill be illustrated with reference to a call option on a stock that doesnot pay dividends (which may be thought of as a portfolio of the stockand its accumulated dividends, re-invested in the stock). It is to beunderstood, however, that the technology is more generally applicable toother options, financial instruments, and trading strategies.

A. Overview of Disclosed Embodiments for Generating Trading Strategiesand Probability Distributions

In certain embodiments disclosed herein, two sets of cleaned price dataare used: first, historical price data for the underlying instrument,including the current price, comprising a high frequency time series(such as daily closing prices); and second, current contemporaneousprice data for the plurality of options, together with the correspondingstrike price and maturity (expiration date) for each option, comprisingoptions of discrete strike prices at each of discrete times to maturity.Further, the contemporaneous price data for the options, which may notbe exactly contemporaneous (e.g., differing as to time of day), can beaccompanied by corresponding (paired) price data for the underlyinginstrument, each corresponding price pair being contemporaneous, ornearly so (e.g., within a time threshold, such as within a minute, 5minutes, an hour, or any other suitable threshold). The data for theunderlying instrument can pertain to some historical interval from astarting date to the current date, and to some maximum frequency (suchas daily). The data for the options can be grouped by each maturity T,and each such group can comprise options with a discrete set of strikeprices. In addition, a contemporaneous present value discount factor canbe provided for each maturity T derived from, for example, a forwardprice for the underlying instrument or a risk-free interest rate.

In some embodiments, a probability distribution model is defined by adistribution parameter function φ, which constitutes one or moreparameters, depending on the form of the function.

In certain embodiments, a short time interval Δt that separateshistorical price data can then be selected, and the underlyinginstrument data can determine a buy-and-hold returns density atfrequency 1/Δt (such as a daily returns density), where the integrateddensity (the cumulated density) corresponds to the historicalprobability distribution of returns. The returns density can beexpressed, for example, in percent per unit time (e.g., scaled by 1/Δt)and can be normalized by the number of observations so as to have totalweight of unity. From the returns density, a conditional distribution ofprices P_(Δt) ^(S) ₀ can be calculated, conditional on the current pricewhich is denoted S₀. In particular embodiments, the conditionaldistribution is estimated from a prior distribution by orthogonalseries. Other embodiments can employ financial econometrics known tothose skilled in the art. One particular embodiment, for example, usesthe so-called double-kernel method. Another embodiment uses thestatistical model known as asymmetric GARCH.

In some embodiments, the values of the set of moments {s*_(k)} of theconditional distribution P_(Δt) ^(S) ₀ are computed (e.g., the mean, thevariance, and so on). (In particular embodiments, a prespecified numberof moments is computed.) The trading strategies {R*_(k)} pertaining tothe conditional distribution P_(Δt) ^(S) ₀ can be determined, asexplicit analytic functions that are orthogonal polynomials in theunderlying instrument price, by algebraic computations with the moments{s*_(k)}. In certain embodiments, the trading strategies represent therelative value of position to be taken in the financial instrument,dependent on (as a function of) the instrument's current market price,by trading (e.g., daily or more frequently). (The overall scale of theposition can be user-selected.) The position can be, for example, long(buy) or short (sell short). The strategies can be displayed to theinvestor or other user of the disclosed technology in the form ofelectronically displayed graphs plotting the value of position againstthe market price of the instrument, for each strategy. In particularembodiments, a prespecified number of strategies is determined.

Two sets of coefficients, {a*_(k)} and {b*_(k)}, can also be computedfrom the moments {s*_(k)} and the strategies {R*_(k)}. In someembodiments, the coefficients {b*_(k)} are checked to ensure that noneof them is equal or very nearly equal to zero, and they are adjusted ifnecessary in accordance with a predetermined or user-selected parameter(a small number) ε. The two sets of coefficients (adjusted if necessary)determine a matrix L₀, the generator of the conditional distribution.The eigenvalues {

} of the matrix L₀ can then be computed. In certain embodiments, theeigenvalues are checked to ensure that no two of them are equal or verynearly equal, and they are adjusted if necessary in accordance with apredetermined or user-selected parameter (a small number) η. The matrixL₀ can be adjusted to ensure that the lowest eigenvalue,

, is equal to or nearly equal to zero. The eigenvalues represent the(expected) excess returns on the corresponding trading strategies,expressed as a return (percent per annum) in excess of the cash marketinterest rate. The excess returns can be displayed to the investor orother user of the disclosed technology in the form of an electronicallydisplayed graph plotting the size of the excess return (e.g., ahistogram) against a respective strategy, ordered by increasing riskclass (e.g., by order of trading strategy). Hedging coefficients, suchas delta and gamma, for the trading strategies can also be determined asexplicit functions by related algebraic computations. The hedgingcoefficients represent the change in the value of a position to be madeby an investor when the current market price of the financial instrumentchanges by a small amount. The coefficients can be displayed on adisplay device (e.g., by being plotted against market price of theinstrument). The strategies and hedging coefficients can be determinedregardless of whether or not options are traded on the underlyingfinancial instrument.

In certain embodiments, a maturity T is selected, corresponding to aselected group of option prices. Each maturity and group can be taken inturn or a specific maturity T can be selected (e.g., by a user via agraphical user interface). From the eigenvalues {

} and the strategies {R*_(k)}, a conditional martingale probabilitydistribution of future financial instrument prices at time T (e.g.,implied solely by underlying instrument prices) can be computed. Thisconditional martingale probability distribution is denoted Q_(T) ^(S) ₀or Q_(T) ^(S) ₀ (S). Probability distributions and densities can bedefined in current value terms (over the price S_(T) prevailing at timeT to which the distribution or density pertains). The conditionalmartingale probability distribution Q_(T) ^(S) ₀ (S) can alternativelybe determined from the counterpart empirical probability distribution,denoted P_(T) ^(S) ₀ (S), which itself may be derived from the set ofhistorical prices for the underlying instrument, by the method of theEsscher transform (exponential tilting).

In some embodiments, and in a similar manner to earlier computationswith P_(Δt) ^(S) ₀ , the values of the set of moments {s_(k)} of theconditional martingale distribution Q_(T) ^(S) ₀ are computed. (Inembodiments, a prespecified number of moments can be computed.) Theorthogonal polynomials {R_(k)} pertaining to the maturity date T can bedetermined, as explicit analytic functions in the underlying instrumentprice, by algebraic computations with the moments. Two sets ofcoefficients, {a_(k)} and {b_(k)}, can be computed, associated to thepolynomials {R_(k)} and the moments {s_(k)}, and the coefficients{b_(k)} can be adjusted if necessary to be non-zero. Also, an auxiliaryset of orthogonal polynomials can be determined as explicit analyticfunctions by algebraic computations with the moments {s_(k)} and thepolynomials {R_(k)}.

In certain embodiments, a two-by-two transformation matrix, called theNevanlinna matrix, is determined by algebraic computations with thecoefficients {b_(k)}, the orthogonal polynomials {R_(k)}, and theauxiliary set of polynomials. Each element of the transformation matrixcan be determined as an explicit analytic function that is aquasiorthogonal polynomial (as explained later).

In some embodiments, the Stieltjes transform

[Q] (an integral transform) of the sought-after distribution Q (thedistribution that best implies the given option prices, consistent withhistorical financial instrument prices), can be determined by thedistribution parameter function φ, through the transformation matrix(the Nevanlinna matrix). The relation between

[Q] and φ is that

[Q] is a linear fractional transformation of φ (a fraction correspondingto the ratio of two linear transformations of φ). By means of an inverseStieltjes transform, the measure Q can be computed from the functionalvalue of the distribution parameter function φ (the values of thefunction for each value of its argument), if the latter is known orspecified. In these embodiments, the distribution parameter function φbelongs to a certain class of functions, namely the compactifiedNevanlinna class. Consequently, the distribution parameter function φ ischaracterized by two extremal values, a maximum and a minimum.

In certain embodiments, the extremal measures (distributions) Q_(F) andQ_(K), respectively called the Friedrichs and Krein measures, aredetermined by means of an inverse Stieltjes transform from the extremalvalues of the distribution parameter function φ. Q_(F) and Q_(K)correspond respectively to the optimal upper and lower bounds on optionprices, and can be used to compute those bounds, when the contractualspecification of an option is given. Using embodiments of the disclosedtechnology, the upper and lower bounds on option prices can bedetermined regardless of whether or not options are traded on theunderlying financial instrument.

In some embodiments, an initial value τ₀ of the distribution parameterfunction is determined, subject to the initial (transient) requirementthat φ take the form of a constant scalar value τ, by optimization inaccordance with the selected group of option prices corresponding to theselected maturity T. Optimization can determine the optimal value τ₀ ofthe scalar τ by selecting a series of test values of τ, and for eachtest value computing the implied distribution by means of an inverseStieltjes transform, computing the consequent implied option prices, andselecting the value τ₀ that implies option prices that best approximatethe group of option prices. In this way, an initial estimate of themartingale distribution Q can be determined, corresponding to τ₀. Thedistribution Q pertains to the selected maturity T, and is sometimesdenoted by Q_(T) to emphasize the connection. Additionally, a set of ninitial values {τ_(i)}_(i=1) ^(n) of the distribution parameter functionφ, where n denotes the number of option prices in the selected group,can be determined by the procedure just described, except that theoptimization is in accordance with a single option price in the selectedgroup, each price being taken in turn.

In certain embodiments, if the set of option prices of selected maturityT is not contemporaneous (within the set), the model can be calibratedbased upon the set of contemporaneous underlying instrument prices. Insome embodiments, an initial value of another matrix (the value of eachmatrix element), an interpolation data matrix called the Pick matrix, isdetermined in accordance with the selected group of options data and theinitial values {τ_(i)}₁₌₁ ^(n), given the underlying instrument data.The Pick matrix expresses the gradient of the distribution parameterfunction φ with respect to the strike price K. It is a matrix ofdimension n-by-n and its off-diagonal elements are equal to differencesin the τ_(i) taken in pairs, each divided by the difference in thecorresponding option strike prices. The diagonal elements are determinedby each τ_(i) in turn by a formula expressing an upper bound to acertain combination of weights in the distribution that is sought after.

In certain embodiments, the initial value of yet another matrix, anauxiliary two-by-two transformation matrix called the Potapov matrix, isdetermined in accordance with the initial value of the Pick matrix, theinitial values {τ_(i)}_(i=1) ^(n), and the n values of the strike pricesof options in the selected group. The possible values of thedistribution parameter function φ are now constrained, through thePotapov matrix, by an auxiliary distribution parameter function U,called the interpolation parameter function or, in brief, theinterpolation function. The interpolation function U also belongs to thecompactified Nevanlinna class of functions. The relation between φ and Uis that φ is a linear fractional transformation of U. Various techniquescan be used to determine an initial and subsequent values of theinterpolation function U. In one embodiment, for instance, a requirementis imposed that the distribution parameter function φ belongs to acertain minimal sub-class of functions consistent with n, the number ofprices provided in the selected group of options data. (The initialrequirement that φ be a constant scalar can be dropped at this stage.)As a consequence, the interpolation function U can take the functionalform of a constant scalar value, which is also denoted by U. An initialvalue U₀ of U can be determined by setting U₀=τ₀, the initial value ofthe distribution parameter function φ.

In some embodiments, in an iterative process, a second value of thedistribution parameter function φ, now a functional value denoted φ_(U)₀ , is determined as an explicit analytic function by the initial valueof the Potapov matrix and the initial value of the interpolationfunction. A second estimate of the distribution Q can be determined bymeans of an inverse Stieltjes transform from the second value of thedistribution parameter function φ, taken as a function (together withthe value, now fixed, of the Nevanlinna matrix). The implied values ofthe prices of the options in the selected group can be computed from theestimate of the distribution. Additionally, a set of constant scalartest values {U_(i)} of U are selected that lie close to U₀ and may fallin a similar range to that spanned by the maximum and minimum of theinitial values {τ_(i)}_(i=1) ^(n). From each value of U_(i), adistribution parameter function φ_(U) _(i) can be calculated, thecorresponding distribution Q_(i) can be determined by means of inverseStieltjes transform, and a corresponding set of implied values of theprices of the options in the selected group can be computed.

Various techniques can be used to determine a final estimate of thedistribution Q. In one embodiment, for example, the final estimate istaken to be the distribution Q_(i) that implies option prices that bestapproximate the selected group of option prices. In another embodiment,a second value of the Pick matrix is constructed before the finalestimate is determined. Accompanying the final estimate, extremalmeasures Q_(H) and Q_(L) are calculated from the extremal values of U,corresponding respectively to conditional upper and lower bounds onoption prices, which are conditional on the option price data.

The preceding method acts, onwards from the selection of a maturity Tcorresponding to a group of option prices, can be repeated for eachgroup of options data, thereby determining distributions {Q_(T)}pertaining to each of the option maturities {T}. In one embodiment, forexample, the final (scalar) values {U_(T)} of the interpolation functionpertaining to the maturities {T} are smoothed as a function of maturity,and the smoothed values are used at each maturity to determine asmoothed set of distributions

that characterize the options market taken as a whole, with associatedimplied smoothed option prices.

Graphical displays can be generated and displayed on a suitable displaydevice (e.g., a computer monitor, tablet display, smart phone display,or other display device) for each of the lower order trading strategies,optionally accompanied by one or more of the respective delta, gamma,and/or other hedging coefficients; and, at each of the selectedmaturities {T}, the display can include each of the martingaledistributions {Q_(T)}, optionally accompanied by the respective optimalupper and lower bounds to the distribution and, optionally, theconditional upper and lower bounds. In certain embodiments, the displayscan be plotted against, for example, the current market price of theunderlying financial instrument.

B. Theoretical Foundations

The theoretical foundation for at least some of the embodimentsdescribed above is based on transforming empirically observed optionprice data into martingale probability distributions through a certaininverse integral transformation, given historical financial instrumentprice data. Under the assumption of no arbitrage in incomplete markets,there exists a set of martingale probability distributions {Q_(T)(S)}that expresses market expectations regarding the potential future prices{S} at time T of a financial instrument. Each Q_(T)(S) incorporates arisk adjustment and, therefore, differs from the probabilitydistribution P_(T)(S). When a plurality of options on the underlyinginstrument are traded, of expiration date T and various strike prices,the observed transaction prices, together with P_(T), provideinformation on Q_(T) in the form of

[Q_(T)], the Stieltjes transform of Q_(T).

[Q_(T)] is an integral transformation (a kind of weighted average ofQ_(T)(S) over all prices S). By means of the inverse Stieltjes transform(an inverse integral transformation), the values of each probabilitydistribution Q_(T) can be recovered from the option prices. The set ofall {Q_(T)} is determined by the set of all possible distributionparameter functions {φ} (a non-denumerably infinite set of cardinalitythe continuum). The short-run conditional probability distributionP_(Δt)(S) can be used to determine a set of uncorrelated tradingstrategies that provide both excess returns commensurate with apredetermined risk class in an investment portfolio, and optimal hedgingin an options portfolio.

More specifically, the starting point for certain embodiments of thedisclosed technology is that, assuming no arbitrage, market expectationsnecessarily take the form of a present value operator that is aself-adjoint extension of the empirically observed operator. Tradingstrategies are eigenfunctions, necessarily orthogonal, of theself-adjoint extensions. In general, there is an infinite set ofpossible self-adjoint extensions, corresponding to the incomplete natureof financial markets. In certain cases, the mathematical theory ofself-adjoint extensions of operators coincides with the classical theoryof moments. The theory of moments describes the set of all possibleprobability distributions for which each moment s_(k) (in an infinity ofmoments s₀, s₁, s₂, . . . ) equals a prescribed value (e.g., empiricallyprescribed). In some embodiments of the disclosed technology, the set ofpossible probability distributions is determined by the so-calledNevanlinna parametrization, in accordance with the Nevanlinna matrix(which can be used to encapsulate the underlying instrument data, asdiscussed below) and the distribution parameter function φ, through aStieltjes transform. To arrive at one distribution out of the many, onecan construct a so-called Nevanlinna-Pick interpolation problem with aPick matrix determined in accordance with the options price data.Finally, any such distribution can be approximated arbitrarily closelyby one of the so-called m-canonical solutions, where m is any positiveinteger. An m-canonical solution is characterized by m free parameters(along with m+1 other parameters which may be considered fixed in thecurrent context). For that reason, embodiments of the disclosedtechnology select an n-canonical solution, where n is the number ofprice observations (the number of options) in the selected group ofoptions data.

C. Detailed Description of Embodiments for Generating Trading Strategiesand Probability Distributions

1. Generating Trading Strategies, Associated Hedging Coefficients, andExpected Excess Returns Using a Probability Distribution Model

FIGS. 2A-2B show a flowchart 200 of an exemplary method for creating atrading strategy, and associated hedging coefficients, in accordancewith an embodiment of the disclosed technology. In particular, theexemplary method in FIGS. 2A-2B is a method for generating tradingstrategies, associated hedging coefficients, and expected excess returnsusing a short-run probability distribution model. The exemplary methodillustrated in FIGS. 2A-2B should not be construed as limiting, however,as any one or more of the illustrated acts can be performed alone or invarious combinations and subcombinations with other method acts.

At 210, historical price data for the underlying financial instrumentare received (e.g., buffered into memory or otherwise input and preparedfor further use). The underlying instrument may be a stock, or basket ofstocks, or stock index, or taken from some other asset class, includinginterest rates, currency exchange rates, commodity prices and creditspreads (or may relate to other areas of economic importance, such asenergy, weather and catastrophe). In particular, as noted earlier,embodiments of the disclosed technology can be applied to a singleasset, a single asset class, such as a market index, and/or a fixedweight portfolio, such as a basket of assets. The data may be alreadyavailable to traders, portfolio managers, or investment banks that wishto employ embodiments of the disclosed technology (e.g., data stored incomputer data storage areas). Alternatively, the data can be obtainedfrom commercially available sources, including electronic disseminatorsof information (such as Bloomberg or Reuters), securities exchanges, andother data vendors. In certain desirable embodiments, price datapertains to actual trades or to indicative quotes or prices marked bytraders (e.g., certain over-the-counter markets). In certainembodiments, the historical price data comprises a high frequency timeseries, meaning prices of at least daily frequency and possiblyintra-day frequency (e.g. daily closing prices in the form of bid-askspread), typically accompanied by the day's high/low prices. (Dailyhigh/low prices are not typically used in an explicit mechanistic way,but can be used in a qualitative way by those skilled in the art toassess characteristics of the data (e.g., the reliability of the data).)When prices are provided in the form of a bid-ask spread, in someembodiments of the disclosed technology, the average of the bid and askprices (or possibly some other derived quantity) is taken to be theprice for the purposes of analysis. The time series can include, forexample, the current price S₀ meaning, for example, the previous day'sclosing price, or the prevailing price in the market today within thelast hour, or a price that prevailed in recent weeks or within anotherselected time period that is of interest for analysis. Thecontemporaneous price data can comprise, for example, a set of currentor recent option prices {C_(i)}, grouped by each maturity (theexpiration date) T, pertaining to a set of strike prices {K_(i)}. (Ifdata is provided in the form of implied volatilities, they can beconverted to prices.) The option prices may not be exactlycontemporaneous (e.g. they may differ by one hour or other time intervalduring the trading day), but are desirably accompanied by pairwisecontemporaneous current prices {S_(i)} for the underlying instrument.The time series current price S₀ for the underlying instrument can betaken to be one of these pairwise contemporaneous current prices{S_(i)}. The prices can be provided in the form of a bid-ask spread, andthe average of the bid and ask prices (or possibly some other derivedquantity) can be taken to be the price in certain embodiments of thedisclosed technology. In some embodiments, the data is supplemented by areal time datafeed (e.g. through an electronic data platform). Any ofthe time frequency or the overall length of the time series can beuser-selected (via a graphical user interface) or predetermined.

In some embodiments, the data is provided in “cleaned” form. Forexample, the data can be filtered for erroneous and suspect or otherwiseunsuitable prices, or can be cleaned by the trader, portfolio manager,investment bank, or other such entity, or by a commercial companyskilled in such cleaning (such as Olsen Ltd.). Preferably, the data iscleaned according to a predetermined or user-selected set of guidelines.Simple examples of cleaning include, for time series, the removal ofprices that deviate from the median of preceding and followingneighbouring prices by more than a predetermined or user-selectedamount. Such cleaning can be desirable for intra-daily data. Adiscussion of suitable data cleaning procedures that can be used withembodiments of the disclosed technology and the construction ofdatabases is given in the book Dacorogna et al., An Introduction to HighFrequency Finance, Chapter 4 (2001). The degree of cleaning and thefrequency of the data provided will depend on, among other factors, thetrading, hedging and pricing objectives of the traders, portfoliomanagers, investment banks, or other such entities who employembodiments of the disclosed technology.

At 220, the distribution of short-run (namely, time horizon Δt) futureprices of the underlying instrument conditional on the current price S₀,denoted by P_(Δt) ^(S) ₀ (S), is determined by the cleaned historicalprice data for the underlying financial instrument. According to oneparticular embodiment of the disclosed technology, the data points inthe time series of historical price data are assumed equally spaced intime, though this need not be the case in actual implementations of thedisclosed technology. In one exemplary embodiment, for instance, theconditional price distribution P_(Δt) ^(S) ₀ (S) (also referred to asthe short-run probability distribution) is computed from every possiblepair of price observations in the time series separated by the selectedtime interval. For example, in this particular embodiment, if there are1,000 consecutive daily price observations, and the selected timeinterval is daily, the number of returns calculated is 999; if theinterval is 2 days, the number is 998. Accordingly, in this embodiment,“short-run” refers to the time horizon Δt, which is user selected orpre-determined.

A variety of methods can be used to compute the short-run probabilitydistribution P_(Δt) ^(S) ₀ (S). One exemplary method using a Whittleestimator is illustrated in FIGS. 1A-1B and discussed below in SectionIII.C.2. In an alternative embodiment, the conditional distributionP_(Δt) ^(S) ₀ (S) is estimated by the double-kernel method (see, e.g.,J. Fan and Q. Yao, Nonlinear Time Series, section 6.5 (2005)). Inanother embodiment, the conditional distribution P_(Δt) ^(S) ₀ (S) isestimated by asymmetric GARCH (see, e.g., J. V. Rosenberg and R. F.Engle, “Empirical Pricing Kernels” in Journal of Financial Economics,vol. 64, pages 341-372 (2002)). Other alternative embodiments includetruncating the time series to a shorter time horizon (possiblyaccompanied by selecting a higher returns frequency 1/Δt), and usingtime weighted prices.

In the embodiment illustrated in FIGS. 2A-2B, the trading strategies{R*_(k)}, the associated hedging coefficients delta, {Δ_(k)}, and gamma,{Γ_(k)}, and the expected excess returns (above the cash interest rate){

} are determined by the conditional distribution P_(Δt) ^(S) ₀ (S),through the moments of the distribution. The strategies, hedgingcoefficients, and expected excess returns can be determined, forexample, regardless of whether or not options are traded on theunderlying financial instrument (as noted earlier). At the outset, atheoretical observation about the illustrated embodiment is that, whileit is desirable that the strategies {R*_(k)} relate to the martingaledistributions {Q} derived from the empirically observed distributionP_(Δt) ^(S) ₀ (S), it turns out that the strategies are in fact the sameset for the martingale distributions because they are related to asingle empirical approximation to the expectations generator (describedbelow), implying that only one set of {R*_(k)} need be calculated. Thenthe values of the set of moments {s*_(k)} of the conditionaldistribution P_(Δt) ^(S) ₀ are computed, in accordance with the usualdefinition for s*_(k), the k-th moment:

s* _(k)=∫₀ ^(∞) S ^(k) P _(Δt) ^(S) ₀ (dS), k=0,1,2, . . . .

In the illustrated embodiment, the computation proceeds as follows. At230, an integer m (non-negative) is selected so as to determine thenumber of moments to be computed. (In certain embodiments, the number mcan be in agreement with the integer m selected during the computationof the conditional price distribution P_(Δt) ^(S) ₀ . For example, thenumber m can be the same as the number m discussed below with respect toFIGS. 1A-1B.). The selection of m can be based on, among other factors,experience with the speed at which the corresponding {a*_(k)} and{b*_(k)} coefficients, described below, converge to approximately fixedvalues.

At 232, the first 2m+2 moments {s*_(k)}₀ ^(2m+1) are computed, includingthe zeroth moment s*₀ which is set equal to unity, thereby normalizingthe probability distribution to unity.

At 240, the trading strategies {R*_(k)} are determined as orthogonalpolynomials in the underlying instrument price by the followingformulae:

$\begin{matrix}{{{R_{k}^{*}(S)} = {\frac{1}{\sqrt{D_{k - 1}D_{k}}}{\begin{matrix}s_{0}^{*} & s_{1}^{*} & \ldots & s_{k}^{*} \\s_{1}^{*} & s_{2}^{*} & \ldots & s_{k + 1}^{*} \\\ldots & \ldots & \ldots & \ldots \\s_{k - 1}^{*} & s_{k}^{*} & \ldots & s_{{2k} - 1}^{*} \\1 & S & \ldots & S^{k}\end{matrix}}}},{k = 0},1,2,\ldots} & {{Equation}\mspace{14mu} 1}\end{matrix}$

where the determinant D_(k) is defined by

$\begin{matrix}{{D_{k} = {{{\begin{matrix}s_{0}^{*} & s_{1}^{*} & \ldots & s_{k}^{*} \\s_{1}^{*} & s_{2}^{*} & \ldots & s_{k + 1}^{*} \\\ldots & \ldots & \ldots & \ldots \\s_{k}^{*} & s_{k + 1}^{*} & \ldots & s_{2k}^{*}\end{matrix}}\mspace{14mu} k} = 0}},1,2,\ldots} & {{Equation}\mspace{14mu} 2}\end{matrix}$

and, by convention, D⁻¹=1.

In particular embodiments, m+1 strategies {R*_(k)}₀ ^(m) are calculated,including the zeroth order R*₀(S)≡1, utilizing all but one of thecalculated moments. It would be possible to calculate the strategiesdirectly from the preceding formulae, taking care to neutralize theaccumulation of rounding errors in large determinants. In oneparticularly desirable implementation, however, the strategies arecalculated recursively. For example, and as shown at 242, the strategiesare calculated recursively by means of the procedure of Schmidtorthogonalization of the sequence of polynomials {S^(k)}, k=0, 1, 2, . .. , (i.e. 1, S, S², . . . ) with respect to the inner (scalar) product,denoted (,), defined by the moments:

(S ^(i) ,S ^(k))=s* _(i+k)   Equation 3

and extended to be bilinear for any functions of S (e.g.(aS^(i)+bS^(j),S^(k))=as*_(i+k)+bs*_(j+k)). Additionally, the norm∥ƒ(S)∥ of any function ƒ of S is defined by the equation

∥ƒ(S)∥=√{square root over ((ƒ(S),ƒ(S)),)}{square root over((ƒ(S),ƒ(S)),)} e.g. ∥S∥=√{square root over (s* ₂)}.

The Schmidt orthogonalization starts by setting R*₀(S)=S⁰=1. R*₁(S) isconstructed from S by subtracting from it a component proportional to 1,such that the result is orthogonal to R*₀(S), and is normalized (i.e.normalized to unity). Hence, it is defined by

${R_{1}^{*}(S)} = \frac{S - {\left( {S,1} \right)1}}{{S - {\left( {S,1} \right)1}}}$

In similar fashion, R*₂(S) is constructed from S² by subtracting from itcomponents proportional to 1 and S, such that the result is orthogonalto R*₀(S) and R*₁(S), and is normalized. Hence, it is defined by

${R_{2}^{*}(S)} = \frac{S^{2} - {\left( {S^{2},1} \right)1} - {\left( {S^{2},{R_{1}^{*}(S)}} \right){R_{1}^{*}(S)}}}{{S^{2} - {\left( {S^{2},1} \right)1} - {\left( {S^{2},{R_{1}^{*}(S)}} \right){R_{1}^{*}(S)}}}}$

Subsequent R*_(k)(S) are constructed in the same way (to be orthogonalto all R*_(i)(S) of lower order, i<k, and to be normalized), as shown at244. Hence, they are defined by

$\begin{matrix}{{R_{k}^{*}(S)} = \frac{S^{k} - {\sum\limits_{i = 0}^{k - 1}{\left( {S^{k},{R_{i}^{*}(S)}} \right){R_{i}^{*}(S)}}}}{{S^{k} - {\sum\limits_{i = 0}^{k - 1}{\left( {S^{k},{R_{i}^{*}(S)}} \right){R_{i}^{*}(S)}}}}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

As constructed, the set {R*_(k)} is normalized to unity. For purposes ofuse of the {R*_(k)(S)} in trading and in graphical depiction, othernormalization conventions may be adopted, e.g. the form of monicpolynomials in which the coefficient of the highest power of price S (orof price scaled by a fixed constant such as √{square root over (s*₂)})is set equal to unity. The monic polynomials, or otherwise normalizedpolynomials, represent the relative value of trading positions,according to a certain normalized scale. They constitute a set oftrading strategies that can be displayed to the user through a graphicalinterface in the form of a graph of each polynomial plotted againstmarket price of the underlying financial instrument.

At 246, and as needed, the procedure of orthogonalization may besupplemented by re-orthogonalization because of numerical cancellationsthat may occur in subtracting components. An exemplary approach tore-orthogonalization that can be used is described in Wilkinson, J. H.,The Algebraic Eigenvalue Problem, chapter 6, sections 32-33 (1988). Itcan be observed that the strategies {R*_(k)(S)} are orthogonal to eachother with respect to any measure (in particular, at the short-run timehorizon) that possesses the given moments, {s*_(k)}₀ ^(2m+1).

At 260, one or more of the strategies are graphically displayed on adisplay device (e.g., each strategy is graphically displayed on adisplay device). FIG. 3 is a representation of a graphical display 300showing one exemplary manner in which the trading strategies can bedisplayed to a user. In FIG. 3, the trading strategies are depicted aspolynomial functions of the price of the underlying financial instrumentindicating the current strategy value. In the illustrated embodiment,each trading strategy is calculated as an explicit polynomial functionof the underlying instrument market price that indicates the value ofthe strategy (investment position) for any given value of the instrumentprice. As the instrument price changes in the market, the tradingstrategy continually adjusts the investment position through dynamictrading so as to maintain the indicated value. The investment position(the quantity of each asset held) comprises a combination of both cashand the underlying instrument. The amounts of each are indicated by thepolynomial function as follows. Negative values indicate borrowedpositions (positions sold short). The constant term in the polynomialfunction corresponds to the amount of cash held. The remaining termsdivided by the instrument price correspond to the amount of theunderlying instrument held. Accordingly, the change in investmentposition necessitated by any given price change can be inferred from thepolynomial function. The change is given in explicit terms by thehedging coefficients, as discussed below.

Hedging coefficients can also be computed and displayed. For example, at250, the delta and gamma hedging coefficients of the trading strategyare determined as analytic functions of the underlying instrument priceS. In particular implementations, the hedging coefficients delta, Δ_(k),and gamma, Γ_(k), for the strategy R*_(k) are the first and secondderivatives respectively, and hence are determined in explicitanalytical form by the following equations:

$\begin{matrix}{\Delta_{k} = \frac{{R_{k}^{*}(S)}}{S}} & {{Equation}\mspace{14mu} 5} \\{and} & \; \\{\Gamma_{k} = \frac{^{2}{R_{k}^{*}(S)}}{S^{2}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

The delta and gamma hedging coefficients can be graphically displayed at260 along with or separate from the graphical display of the strategies.

At 270, one or more additional hedging coefficients are optionallycomputed. For example, other hedging coefficients can be derived insimilar fashion (e.g., the third derivative is related to skewness ofthe distribution).

In general, the delta hedging coefficient expresses the change in valueof the trading strategy (investment position) for a small change in theprice of the underlying instrument. In other words, it equals the rateof change of value with respect to price. For a small change in price,it indicates the required change in investment position in order tomaintain the investment strategy as follows. Delta is expressed as apolynomial function of price. Excluding the constant term in thepolynomial and dividing the remaining terms by price (of the underlyinginstrument) equals the rate of change (with respect to price) of theinvestment position in the underlying instrument. The gamma hedgingcoefficient, known to those skilled in the art, similarly expresseschanges in the investment position additional to those indicated by thedelta coefficient when prices jump in significant increments.

In an alternative embodiment, the trading strategies are calculated inthe same manner as the exemplary embodiment described above, exceptingthat they are calculated with respect to the representative martingaledistribution (described below) at a selected maturity T, not theshort-run distribution. The computations of hedging coefficients andexpected excess returns, described below, also proceed in the samemanner, excepting that they are based on the representative martingaledistribution.

At 280, the expected excess returns on the strategies {R*_(k)(S)}₀ ^(m)are computed. For example, the expectations generator L₀, and itseigenvalues {

}, which equal the expected excess returns of the corresponding tradingstrategies, and the {a*_(k)}, and {b*_(k)} coefficients can be derivedfrom the strategies {R*_(k)}₀ ^(m) and the moments {s*_(k)}₀ ^(2m+1).The numerical calculation of the {R*_(k)(S)} by Schmidtorthogonalization (described above) gives rise to a second order finitedifference equation

b* _(k−1) R* _(k−1) +a* _(k) R* _(k) +b* _(k) R* _(k+1) =SR* _(k) ,k=1,2, 3,   (Equation 7)

with the initial condition

(a* ₀ −S)1+b* ₀ R* ₁=0,

in other words, the condition R*₀=1 (and implicitly b*⁻¹=0). Each pairof coefficients a*_(k) and b*_(k−1) can be determined by the precedingequation (1), which generates R*_(k+1). Alternatively, the coefficientscan be computed from the explicit expressions

$\begin{matrix}{{a_{k}^{*} = {{G\left\{ {{{SR}_{k}^{*}(S)}{R_{k}^{*}(S)}} \right\} \mspace{14mu} b_{k}^{*}} = \frac{\sqrt{D_{k - 1}D_{k + 1}}}{D_{k}}}},{k = 0},1,2,\ldots} & {{Equation}\mspace{14mu} 8}\end{matrix}$

taking care to neutralize the accumulation of rounding errors in largedeterminants, where the functional G{R(S)} is defined by

G{R(S)}=p ₀ s* ₀ +p ₁ s* ₁ + . . . +p _(n) s* _(n)

for R(S)=p₀+p₁S+ . . . +p_(n)S^(n). From these expressions, it can beseen that the computation of the coefficient a*_(m) requires knowledgeof the moments {s*_(k)}₀ ^(2m+1). Accordingly, from the set of moments{s*_(k)}₀ ^(2m+1) the set of coefficients {a*_(k)}₀ ^(m) and {b*_(k)}₀^(m−1) can be determined, and this is the set of coefficients that iscomputed in certain embodiments of the disclosed technology. Thecoefficients {b*_(k)} can be checked to ensure that none of them isequal or very nearly equal to zero, and they can be adjusted ifnecessary to be strictly positive in accordance with a predetermined oruser-selected parameter (a small number) ε. An exemplary method ofadjustment that can be used in embodiments of the disclosed technologyis described by J. H. Wilkinson, The Algebraic Eigenvalue Problem,Chapter 5, sections 45-46 (1988). The two sets of coefficients (adjustedif necessary) determine a symmetric tridiagonal matrix L₀, the generatorof the conditional distribution P_(Δt) ^(S) ₀ (S), identified by theequation

$\begin{matrix}{L_{0} = \begin{pmatrix}a_{0}^{*} & b_{0}^{*} & 0 & 0 & 0 & \ldots & \ldots \\b_{0}^{*} & a_{1}^{*} & b_{1}^{*} & 0 & 0 & \ldots & \ldots \\0 & b_{1}^{*} & a_{2}^{*} & b_{2}^{*} & 0 & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ddots & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & a_{m - 1}^{*} & b_{m - 1}^{*} \\\ldots & \ldots & \ldots & \ldots & \ldots & b_{m - 1}^{*} & a_{m}^{*}\end{pmatrix}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

where the {a*_(k)} are real numbers, and the {b*_(k)} are strictlypositive. (In analytical terms, with an infinite set of moments andcorresponding infinite set of coefficients, the expectations generatorL₀ is an infinite matrix. L₀, which encapsulates the empirical moments,represents the empirical approximation to its self-adjoint extensions,which act on differing densely-defined domains.)

The eigenvalues {

}₀ ^(m) of the matrix L₀ can be computed, for example by the bisectionmethod. See, e.g., J. H. Wilkinson, The Algebraic Eigenvalue Problem,Chapter 5, sections 36-43 (1988). The eigenvalues can be checked toensure that no two of them are equal or very nearly equal, and they canbe adjusted if necessary in accordance with a predetermined oruser-selected parameter (a small number) η.

An exemplary method of adjustment that can be used in embodiments of thedisclosed technology is described in J. H. Wilkinson, The AlgebraicEigenvalue Problem, Chapter 5, section 59 (1988)). In an alternativeembodiment, one could work with sub-matrices and calculate correspondingeigenvalues and eigenvectors. One could also work with a generator thatis a function of the underlying instrument price, where L₀=L₀(S). Thematrix L₀ can be adjusted so as to help ensure that the lowesteigenvalue,

, is zero. If the unadjusted

equals αa, for example, then the adjustment can be to subtract theconstant α (α times the unit matrix) from L₀, which shifts alleigenvalues equally. This exemplary adjustment corresponds to takingzero as the baseline for excess returns over the risk-free interestrate.

By way of explanation, the theoretical reason for adjusting the {b*_(k)}coefficients to be strictly positive and the separation of theeigenvalues {

} (the latter implies the former, but not conversely) is as follows.Under the assumption of no arbitrage, the expectations generator matrixL₀ is of finite multiplicity for financial instrument price processesthat are, for example, time-homogeneous Markov processes, though notnecessarily continuous (prices may jump). That means that eacheigenvalue

is associated with a finite number of eigenvectors. For a single asset,single asset class, or fixed weight portfolio or basket, representing asingle risk factor (possibly, a fixed weight combination of multiplefactors), the expectations generator is of single multiplicity (on anappropriately reduced state space defined by a cyclic vector). That is,if the state is defined by the configuration of asset prices (asdistinct from Arrow-Debreu states), then the multiplicity is single.Then each eigenvalue

is associated to a single eigenvector (no eigenvalues coincide). Themethods of classical moment theory proceed from the basis that thegenerator matrix L₀ is of single multiplicity. In practice, expectationsgenerators of higher than single multiplicity may behave empiricallylike the single multiplicity case for aggregate portfolio values: theempirical data can be adjusted by empirically insignificant amounts toarrive at a generator matrix L₀ of single multiplicity. This possibilityis another justification for treating L₀ as a matrix of singlemultiplicity.

2. Generating Conditional Probability Distributions Using the WhittleEstimator

FIGS. 1A-1B show a flowchart 100 of an exemplary technique fordetermining a conditional returns distribution by an orthogonal seriesas may be used at method act 220 of FIGS. 2A-2B. The exemplary methodillustrated in FIGS. 2A-2B should not be construed as limiting, however,as any one or more of the illustrated acts can be performed alone or invarious combinations and subcombinations with other method acts.

At 110, a returns frequency 1/Δt is received (e.g., daily frequency), orequivalently a time interval Δt is received, where Δt defines the timeinterval that will be used to generate pairs of time series data pointsseparated by the interval, so as to calculate returns. The returnsfrequency can be predetermined or can be received (e.g., buffered orotherwise input and prepared for use) as part of user data generatedthrough a graphic user interface or other interface with a user. In oneparticular implementation, the minimum possible value of the timeinterval (corresponding to the maximum frequency) is the minimuminterval in the time series; other intervals are integral multiples ofthe minimum.

Then, in one particular embodiment of the disclosed technology, thereturns {r_(j)} over the intervals Δt are computed to be equal to thepercentage change, expressed in percent per unit time, in successivepairs of prices that are separated by time interval Δt; and all possiblepairs can be taken. In this embodiment, each return r_(j) is associatedto the initial price S_(j) (the price at the start of the interval Δt)from which it was derived. The set of returns {r_(j)} constitutes areturns density, which can be normalized to have a weight of unity(normalized to be a probability density). The returns density candetermine a conditional price distribution, conditional on the currentprice S₀, namely the probability distribution of all prices generated bythe returns density acting over the time interval Δt starting from fixedinitial price S₀. This distribution can be denoted by P′_(Δt) ^(S) ₀ (S)(to be distinguished from P_(Δt) ^(S) ₀ (S)), and called theunconstrained conditional price distribution (or just unconstraineddistribution) at time horizon Δt, where “unconstrained” indicates thatthe entire set of returns {r_(j)} has been used in the computation.

At 120, the unconstrained conditional price distribution P′_(Δt) ^(S) ₀(S) is computed. In the illustrated embodiment, the conditionaldistribution P_(Δt) ^(S) ₀ (S) is estimated by an orthogonal series,namely the orthogonal polynomials {R′_(k)(S)}, described below, asderived from the unconstrained distribution P′_(Δt) ^(S) ₀ (S). In thisembodiment, the distribution P_(Δt) ^(S) ₀ (S) is set equal to theWhittle estimator as follows.

The Whittle estimator, denoted by {circumflex over (P)}_(Δt) ^(S) ₀ (S),is given by the following expression:

$\begin{matrix}{{{{{\hat{P}}_{\Delta \; t}^{S_{0}}(S)} = {{{{\hat{P}}_{\Delta \; t}^{\prime \; S_{0}}(S)}\left\{ {1 + {\sum\limits_{k = 1}^{\infty}{{\hat{\beta}}_{k}{R_{k}^{\prime}(S)}}}} \right\} \mspace{14mu} {truncated}\mspace{14mu} {at}\mspace{14mu} k} = m}},\mspace{20mu} {{\hat{\beta}}_{k} = {\left( \frac{L}{L - 1 + \pi_{k}} \right){\overset{\_}{R}}_{k}^{\prime}}},\mspace{14mu} {{\overset{\_}{R}}_{k}^{\prime} = {L^{- 1}{\sum\limits_{j = 1}^{L}{R_{k}^{\prime}\left( {\overset{\sim}{S}}_{j} \right)}}}}}\mspace{20mu} {{k = 1},2,\ldots \mspace{14mu},}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

where {circumflex over (P)}′_(Δt) ^(S) ₀ (S) denotes the priordistribution, and implicitly the zeroth-order polynomial is the functionequal to 1 (R′₀(S)=1), representing the cash asset of constant price 1.The notation, ̂, denotes a statistical estimator, the dashed notation,′, simply denotes an alternative version and should not be mistaken fordifferentiation, and the tilde notation, ˜, denotes a random variable.Δt denotes the selected time interval as described above, and the{{circumflex over (β)}_(k)} are the Whittle coefficients in theorthogonal series expansion (with implicitly {circumflex over (β)}₀=1for a normalized distribution). In one particular embodiment, theelements of the Whittle estimator are computed as follows.

At 130, the prior distribution (more precisely, its estimator) is setequal to the unconstrained conditional price distribution P′_(Δt) ^(S) ₀(S). Furthermore, the moments {s′_(k)} are computed for thedistribution. In particular implementations, the first 2m+1 moments ofthe prior distribution (including the zeroth order (normalizing) moment)are computed. The integer m can be selected as described below.

At 132, the polynomials {R′_(k)} are calculated in the same manner asare the trading strategies described below, excepting that they arebased on the prior distribution P′_(Δt) ^(S) ₀ (S), and for that reasonthe calculations are not described here. In a particular implementation,the integer m is selected to be the desired order of the highest orderpolynomial to be calculated (R′_(m)), meaning that, in the illustratedembodiment, m+1 polynomials {R′_(k)}₀ ^(m) are calculated, including thezeroth order polynomial R′₀(S)≡1, so as to approximate the distribution.(The notation { }₀ ^(m) denotes the range of values of the variableindex.) The resulting orthogonal polynomials {R′_(k)(S)} satisfy theconditions that R′₀(S) represent the cash asset and the orthogonalpolynomials {R′_(k)(S)}, k=0, 1, 2, . . . , m, are ordered by decreasingsmoothness.

At 140, the bandwidth control parameter L is set. In the illustratedembodiment, the bandwidth control parameter L is set (e.g., via apredetermined selection or via user-selection through a graphical userinterface or other interface with the user) to the desired number oftime series observations at or close to the current price S₀.

At 142, the L observations closest to S₀, whether higher or lower, aretaken. In particular, the subset of n returns that are determined by theselected bandwidth control parameter L are identified from the set ofreturns {r_(j)}, and the bandwidth-constrained conditional pricedistribution P″_(Δt) ^(S) ₀ (S) is constructed in non-parametric form.

At 150, the prior precision parameters are set to be exponentiallyincreasing. The values {π_(k)} are the prior precision parameters, andπ_(k)≧1. The {π_(k)} can be set to be exponentially increasing by, forexample, the following formula: π_(k)=c^(k) where c is a single digitinteger.

At 160, one or more Whittle coefficients {circumflex over (β)}_(k) arecalculated (e.g., each Whittle coefficient) from the averaged polynomialR′_(k), where the averaged polynomial is averaged over thebandwidth-constrained values of price (the L observations closest toS₀). The computation of the Whittle coefficient can be performed inaccordance with the formula given above. In the illustrated embodiment,this computation comprises the following. At 162, the values of eachR′_(k), averaged over the bandwidth-constrained conditional pricedistribution P″_(Δt) ^(S) ₀ (S), are computed. At 164, the correspondingWhittle coefficient {circumflex over (β)}_(k) is computed from theaverage value of R′_(k), the prior precision parameter π_(k), and thebandwidth control parameter. At 166, the Whittle estimator is computedas an orthogonal series modification of the prior distribution P′_(Δt)^(S) ₀ (S).

At 170, the Whittle estimator is evaluated to determine if it gives anon-negative density. If the eventually calculated probability densityis negative, the prior precision parameters can be increased so as toensure that the density is positive.

At 180, the conditional distribution can be set equal to the Whittleestimator:

P _(Δt) ^(S) ₀ (S)={circumflex over (R)} _(Δt) ^(S) ₀ (S)

One advantage of the Whittle estimator is that it is directly applicableto data samples (sets of L bandwidth-constrained prices) of arbitrarysize. Consequently, the Whittle estimator does not rely upon asymptoticproperties. The distinction is meaningful because approaches that relyupon asymptotic optimality of sequences of estimators often invoke anassumption of ergodicity. Ergodicity implies that there is somepossibility that any state of the market (defined by asset prices) couldevolve into any other state, given a long enough time period. In thesecircumstances, the matrix representing the expectations operator isirreducible, which implies that the market is complete. But that is justthe assumption that one wishes to avoid, in order to attain greaterrealism.

3. Generating Martingale Distributions of Future Prices From UnderlyingInstrument Prices

FIG. 4 is a schematic block diagram showing components of and theoverall flow for an exemplary method of generating a set of two or moremartingale probability distributions from price data of an underlyingfinancial instrument. The particular components shown in FIG. 4 shouldnot be construed as limiting, however, as the illustrated components canbe used alone or in various combinations and subcombinations with oneanother. Similarly, the illustrated actions shown in FIG. 4 can beperformed alone or in various combinations and subcombinations with oneanother.

In the illustrated embodiment, a conditional martingale distribution offuture prices at time T (implied solely by underlying instrumentprices), denoted Q_(T) ^(S) ₀ (S), is determined by the short-runconditional empirical distribution P_(Δt) ^(S) ₀ (S). Probabilitydistributions and densities can be defined in current value terms (theydescribe prices prevailing at the time to which the distribution ordensity pertains, as opposed to discounted present value prices).

a. Computing the Representative Distribution Q_(T) ^(S) ₀ (s)

At 410, an option maturity T is received (e.g., buffered into memory orotherwise input and prepared for further use). The option maturity Tcorresponds to the expiration date for a group of one or more options.The option maturity T can be received, for example, from user datagenerated by a user selecting the maturity T via a graphical userinterface or other interface with the user. In particularimplementations, multiple maturities are received and each maturity istaken in turn, though other orders are also possible.

At 411, historical price data for the underlying instrument is alsoreceived. For example, the historical price data can be accessed andinput using any of the methods or formats described above with respectto method act 210.

At 412, the conditional martingale probability distribution Q_(T) ^(S) ₀is computed using a probability distribution model. For example, theconditional martingale probability distribution Q_(T) ^(S) ₀ can becomputed by means of a time evolution of the Fourier-type components ofthe initial distribution P₀(S), in accordance with the expectationsgenerator L₀ and its eigenvalues {

}₀ ^(m), as follows. The Fourier-type components are taken to be theorthogonal functions {R*_(k)(S)}₀ ^(m). Further, the components{R*_(k)(S)}₀ ^(m) can be identified with the eigenvectors of theexpectations generator (in a suitably chosen coordinate basis for thestate space). The coefficients {c_(k)}₀ ^(m) of the components can bedetermined in the usual way by Fourier analysis, namely the integral ofthe product of the distribution P₀(S) and the component concerned withrespect to a distribution with moments {s*_(k)}₀ ^(2m+1):

c _(k)=(R* _(k)(S),P ₀(S)), k=0, 1, 2, . . . , m,

where the integral is written as the scalar product defined earlier inequation (3). If, for example, the initial distribution is a fixed valueof price S₀, then the distribution P₀(S) has a Dirac delta densityδ(S₀). Whatever the distribution, each coefficient can be regarded astime varying and is evolved through time depending upon thecorresponding eigenvalue in accordance with the equation

c _(k)(T)=exp(−

T)c _(k)   Equation 11

where c_(k) is equivalent to time zero c_(k)(0). The distribution Q_(T)^(S) ₀ is then equal to the sum of each function R*_(k)(S) multiplied byits corresponding time-evolved coefficient (Σ₀ ^(m)c_(k)(T)R*_(k)(S)),re-normalized to unity by a suitable constant:

${Q_{T}^{S_{0}}(S)} = {\frac{1}{\int_{0}^{\infty}{{S}\; {\sum\limits_{0}^{m}{{c_{k}(T)}{R_{k}^{*}(S)}}}}}{\sum\limits_{0}^{m}{{c_{k}(T)}{{R_{k}^{*}(S)}.}}}}$

In the foregoing formulae, particular numerical techniques can be usedto approximate the integrals (e.g. truncation at an upper bound toprice).

The distribution Q_(T) ^(S) ₀ can be adjusted, as needed, so that thefirst moment is equal to the forward price S_(T) ^(forward) (equal to S₀in discounted present value terms), by scaling by a constant factorprices over which Q_(T) ^(S) ₀ is defined. A forward price S_(T)^(forward) contemporaneous with the current price S₀ can be availablefor input directly as an empirical datum, or indirectly through adiscount factor. In the latter case, S_(T) ^(forward) is set equal tothe ratio S₀/DF_(T). Here, DF_(T) refers to the discount factor, or“deflator”, pertaining to maturity T. If a forward price of theunderlying financial instrument at date T, denoted S_(T) ^(forward) andcontemporaneous with the current price S₀, is available from observedmarket transactions or quotes, then the discount factor is set equal tothe ratio

${DF}_{T} = {\frac{S_{0}}{S_{T}^{forward}}.}$

Otherwise, the discount factor is set equal to the exponential

DF _(T)=exp(−r _(T) T),

where r_(T) is a risk-free interest rate (e.g. a Treasury yield curvezero coupon rate) pertaining to maturity T, and derived from observedmarket rates contemporaneous with the current price S₀ of the underlyingfinancial instrument. The formula is adapted in the case that theunderlying instrument pays dividends or interest rate coupons.

In some embodiments, additional information can be supplied to users ofthe disclosed technology. For example, an estimate of the empiricalprobability distribution pertaining to time T, conditional on thecurrent price S₀ of the underlying instrument, denoted by P_(T) ^(S) ₀ ,can be provided.

Conversely, if P_(T) ^(S) ₀ (S) is computed by another means, which cancomprise computing P_(T) ^(S) ₀ (S) from the set of historical pricesfor the underlying instrument, then Q_(T) ^(S) ₀ (S) can bealternatively calculated from P_(T) ^(S) ₀ (S) by the method of theEsscher transform. The “exponentially tilted” distribution e^(−αs)P_(T)^(S) ₀ (S) is fitted to the forward price S_(T) ^(forward)) that isdescribed above by fitting α (the first moment of the distribution isset equal to S_(T) ^(forward)). (It is assumed that S_(T) ^(forward) iseither an empirical datum, or derived from an empirically given S₀through a given discount factor.) Denoting the unique fitted value of αby θ, A_(T) ^(S) ₀ (S) is determined by setting Q_(T) ^(S) ₀(S)=e^(−θs)P_(T) ^(S) ₀ (S). The distribution Q_(T) ^(S) ₀ (S) can beregarded as a representative distribution in that any distribution withthe same set of moments can serve the purposes that it serves.

The values of the determinants {D_(k)}₀ ^(m), from equation (2), can bechecked for the presence of any determinant that is zero or nearly zeroLikewise, the values of the determinants {D_(k) ⁺}₀ ^(m) defined by theequations

$\begin{matrix}{{D_{k}^{+} = {{{\begin{matrix}s_{1}^{*} & s_{2}^{*} & \ldots & s_{k + 1}^{*} \\s_{2}^{*} & s_{3}^{*} & \ldots & s_{k + 2}^{*} \\\ldots & \ldots & \ldots & \ldots \\s_{k + 1}^{*} & s_{k + 2}^{*} & \ldots & s_{{2k} + 1}^{*}\end{matrix}}\mspace{14mu} k} = 0}},1,2,\ldots} & {{Equation}\mspace{14mu} 2}\end{matrix}$

can be checked for the presence of any determinant that is zero ornearly zero. If there is any such determinant among either the {D_(k)}₀^(m) or the {D_(k) ⁺}₀ ^(m), it can be concluded that there is only onemartingale measure Q_(T) corresponding to a particular time horizon T(the measure Q_(T) is unique). Such an occurrence is typicallyexceptional. Then Q_(T) can be set equal to the conditional martingaledistribution QT and constitutes the upper and lower bound, theconditional upper and lower bound, and the optimal martingaledistributions at maturity T. Another option maturity can then beselected in an iterative process. For each maturity, and in oneparticular embodiment, the same conclusion is applied that there is onlyone martingale measure, and the unique measure is set equal to theconditional martingale distribution. Otherwise, and in some embodimentsof the disclosed technology, the method proceeds as follows.

b. Computing Extremal Distributions Q_(F) and Q_(K)

A probability distribution model that generates an optimal (“best fit”)martingale probability distribution Q_(T) is determined as to functionalform by the Nevanlinna parametrization, wherein the Nevanlinna matrix isdetermined by the distribution Q_(T) ^(S) ₀ through the moments of thedistribution, the extremal values of the distribution parameter functionφ determine extremal measures Q_(F) and Q_(K), and φ itself isparametrized by the Potapov matrix, which is determined by options pricedata.

The embodiment illustrated in FIG. 4 proceeds as follows. At 420, thehistorical price data is encapsulated in the Nevanlinna matrix. Inparticular implementations, the Nevanlinna matrix is defined by:

$\begin{pmatrix}{A(z)} & {C(z)} \\{B(z)} & {D(z)}\end{pmatrix},$

where A(z), B(z), C(z), and D(z) are four real polynomials in thevariable z, which is a complex number, and by “real” is meant that eachpolynomial has real coefficients. (The polynomial D(z) should not beconfused with the determinants {D_(k)}.) In certain embodiments,approximations to the four polynomials are constructed in the followingway. Orthogonal polynomials {R_(k)(S)}₀ ^(m) are calculated in the samemanner as are the trading strategies described above, excepting thatthey are based on the distribution Q_(T) ^(S) ₀ (S). Additionally, the{b_(k)} coefficients associated to the {R_(k)} are computed in the samemanner as described above (e.g. through Schmidt orthogonalization andre-orthogonalization), including adjustment if needed to make the{b_(k)} strictly positive (and the {a_(k)} coefficients are computed atthe same time). Another set of orthogonal polynomials, auxiliary to the{R_(k)}, is defined by

${W_{k}(S)} = {G_{S^{*}}\left\{ \frac{{R_{k}(S)} - {R_{k}\left( S^{*} \right)}}{S - S^{*}} \right\}}$k = 0, 1, 2, …  m,

where G_(S*) denotes the functional G as defined earlier excepting withrespect to the independent variable S*. W_(k) is a polynomial of degreek−1. The polynomials {W_(k)(S)}₀ ^(m) are computed from the {R_(k)(S)}₀^(m). Then the polynomials A_(n)(z), B_(n)(z), C_(n)(z), D_(n)(z), n=1,2, . . . are defined as below:

A _(n)(z)=b _(n−1) {W _(n−1)(0)W _(n)(z)−W _(n)(0)W _(n−1)(z)},

B _(n)(z)=b _(n−1) {W _(n−1)(0)R _(n)(z)−W _(n)(0)R _(n−1)(z)},

C _(n)(z)=b _(n−1) {R _(n−1)(0)W _(n)(z)−R _(n)(0)W _(n−1)(z)},

D _(n)(z)=b _(n−1) {R _(n−1)(0)R _(n)(z)−R _(n)(0)R _(n−1)(z)}.

Here, the variable z is substituted for the real variable S in thepolynomials {R_(k)} and {W_(k)}. B_(n)(z) and D_(n)(z) arequasiorthogonal polynomials of degree n, and A_(n)(z) and C_(n)(z) arethe corresponding numerators. By “quasiorthogonal polynomial of degreen” is meant a sum of R_(n) and R_(n−1) with fixed coefficients. By “thecorresponding numerator” is meant the corresponding sum of W_(n) andW_(n−1) with the same fixed coefficients (respectively). For n tendingto infinity (n→∞), A_(n)(z), B_(n)(z), C_(n)(z), and D_(n)(z) tenduniformly to their limits for all finite values of z, denotedrespectively by A(z), B(z), C(z), and D(z). It is the latter polynomialsthat constitute the Nevanlinna matrix as defined above. In certainembodiments, for the purposes of calculation, the limits A(z), B(z),C(z), and D(z) are approximated by the m-th order approximants,respectively, A_(m)(z), B_(m)(z), C_(m)(z), and D_(m)(z) (where moments{s_(k)}₀ ^(2m+1) have been computed), and similarly the Nevanlinnamatrix is approximated by the m-th order approximant, defined

$\begin{pmatrix}{A_{m}(z)} & {C_{m}(z)} \\{B_{m}(z)} & {D_{m}(z)}\end{pmatrix}.$

Accordingly, the polynomials A_(m)(z), B_(m)(z), C_(m)(z), and D_(m)(z)are calculated as explicit analytical functions, and the Nevanlinnamatrix (in approximation) is thereby determined.

The Stieltjes transform of the probability distribution Q(S) is definedby

$\begin{matrix}{{\lbrack Q\rbrack} \equiv {\int_{0}^{\infty}{\frac{Q\left( {S} \right)}{S - z}.}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

The lower limit to the integral is zero, because the underlyingfinancial instrument (as illustrated, a stock) is of limited liabilityand, therefore, cannot take a negative value.

can be written as

to emphasize its dependence on the complex number z, and further as

where z=ξ+iη, i denoting √{square root over (−1)} and ξ and ηrespectively denoting the real and imaginary components of z. Allsolutions {Q} to the moment problem (distributions Q that possess theprescribed moments {s_(k)}) and assuming that there is more than onesolution, satisfy the parametric formula (the “Nevanlinnaparametrization”):

$\begin{matrix}{{{\lbrack Q\rbrack} \equiv {\int_{0}^{\infty}\frac{Q\left( {S} \right)}{S - z}}} = {{- \frac{{A(z){\varphi (z)}} - {C(z)}}{{{B(z)}{\varphi (z)}} - {D(z)}}}\mspace{14mu} {Incomplete}\mspace{14mu} {Markets}\mspace{14mu} {Formula}}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

where the distribution parameter function φ belongs to the “compactifiedNevanlinna class” of functions, φ ∈ N (the Nevanlinna class N augmentedby the constant ∞ (which constitutes a compact set)). In this way, theparametric formula determines the Stieltjes transform as a fractionallinear transformation of the distribution parameter function. TheNevanlinna class of functions N (sometimes referred to as Herglotzfunctions) consists of all analytic functions w=ƒ(z) that map thehalf-plane Im z>0 to the half-plane Im w≧0, where Im denotes theimaginary part of a complex number value. (Nevanlinna functions can beconsidered to be extended to the lower half-plane by symmetry.) Theimmediately preceding formula will hereinafter be referred to as the“incomplete markets formula”.

When markets are incomplete, knowledge of even all the moments (aninfinite set) of a martingale probability distribution is insufficientto determine the distribution. As discussed below, a fixed infinite setof moments is consistent with an infinite set of distributions.

The inverse Stieltjes transform is given by the Stieltjes-Perroninversion formula:

$\begin{matrix}{{\frac{{Q\left( {S + 0} \right)} + {Q\left( {S - 0} \right)}}{2} - \frac{{Q\left( {0 +} \right)} + {Q(0)}}{2}} = {\lim\limits_{\eta->0}{\frac{1}{\pi}{\int_{0}^{S}{{Im}\; _{\xi + {i\; \eta}}{\xi}}}}}} & {{Equation}\mspace{14mu} 15}\end{matrix}$

Here, “lim” denotes mathematical limit and +0, −0, and 0+ denote thelimits approached from above, below and above respectively. Ann-canonical solution to the incomplete markets formula is defined to bea distribution Q corresponding to a distribution parameter function φthat is a rational function of n-th degree and is real on the real axis.Because certain embodiments of the disclosed technology require thesolution Q(S) to be an n-canonical solution, where n denotes the numberof options of maturity T, Q(S) is a pure point measure and the weight atprice zero (Q(0+)−Q(0)), if any, can be calculated explicitly as theresidue at the pole (the singularity) at S=0. For the same reason, thepoints of increase of Q(S) can be located explicitly and hence thepoints where there is no increase in Q(S) are known and the values of Qat such points are determined by the Stieltjes-Perron inversion formula.(The number of points of increase are infinite for the exact analyticalsolution, but finite when using the m-th order approximants.)Computations performed in accordance with the preceding formula arereferred to hereinafter as the inverse Stieltjes transform sub-model.

Consequently, the probability distribution model, which is based on theincomplete markets formula, can be used to compute any solution Q(S)that is an n-canonical solution, through appropriate selection of thedistribution parameter function φ. Furthermore, every solution Q of themoment problem can be approximated arbitrarily closely by an n-canonicalsolution, for some value of n (in mathematical terms, the set ofcanonical solutions is dense in the set of all solutions).

In the case that the conjectured Q is determined by φ=τ, a scalarquantity, which in fact corresponds to zeroth-order canonical solutions,the results of Stieltjes-Perron inversion can be determined explicitlyas follows. The location of the poles (the points of positiveprobability weight) are the solutions to the equation:

B(S)τ−D(S)=0;

and the size of the probability weight at such points equals ρ(S) givenby

$\begin{matrix}{{\rho (S)} = \frac{1}{{{D(S)}\frac{{B(S)}}{S}} - {{B(S)}\frac{{D(S)}}{S}}}} & {{Equation}\mspace{14mu} 16}\end{matrix}$

In some embodiments of the disclosed technology, once a martingaleprobability distribution Q_(T) is determined, the price of an option orother derivative is determined in the following manner. For an optionwith payoff f(S_(T)) at time T, where S_(T) denotes the prevailing priceof the underlying financial instrument at time T, the price C of theoption is given by the formula:

C=DF _(T)∫₀ ^(∞i f)(S _(T))Q _(T)(dS _(T))   Equation 17

Here, DF_(T) refers to the discount factor, defined above, pertaining tomaturity T. Computations in accordance with the above formula for priceC are hereinafter referred to as the options pricing sub-model.

At 422, the extremal measures Q_(F) and Q_(K) (respectively, theFriedrichs and Krein measures) are determined by the extremal scalarvalues of the distribution parameter function φ in the incompletemarkets formula. For the case of a measure over positive values [0, ∞),corresponding to a limited liability financial instrument that does nottake negative prices, preferably these values are taken to be φ=−∞ andφ=0 respectively for Q_(F) and Q_(K),

[Q _(F) ]=−A(z)/B(z)

[Q _(K) ]=−C(z)/D(z)   Equation 18

From these equations, the measures Q_(F) and Q_(K) are calculated by theinverse Stieltjes transform sub-model at 424. The measure Q_(F)generates the upper bound to an option price and Q_(K) the lower bound,through the options pricing sub-model. In an alternative embodiment, theabsolute value of φ corresponding to Q_(F) is taken to be large butfinite, and may depend on the margin by which the lowest eigenvalue(before adjustment) of the empirical generator L₀ exceeds zero in astatistically significant way. The upper and lower bounds to an optionprice can be determined regardless of whether or not options are tradedon the underlying financial instrument (as noted earlier, provided thatthe financial instrument is of limited (or approximately limited)liability).

c. Computing a Best Fit of Distribution Parameter Function φ

As described above, and in one embodiment of the disclosed technology,an initial optimal value τ₀ of the distribution parameter function φ isdetermined by setting φ to be a scalar (a function equal to a fixed realnumber) and, from a set of scalar test values, choosing that value thatgives a best fit to the selected group of option prices. At 430, one ormore test values are generated. The one or more test values caninitially be randomly generated, or can be user input, or based on someother criteria (e.g., a heuristic or from past sub-model runs). Given atest value, the implied price for each option in the group is computedby means of the inverse Stieltjes transform sub-model (applied at 432)and the options pricing sub-model (applied at 434).

At 436, the best fit set for the whole set of one or more options can bedetermined by comparing the results to contemporaneous (or nearlycontemporaneous) price data for options. The best fit can be determinedaccording to a predetermined or user-selected criterion (e.g., leastsquares, the fit that minimizes the sum of the squares of the deviationsbetween each computed price and its empirical data counterpart, or othersuch criteria). The best fit determines the value τ₀. At this point, thenumber of options in the selected group at maturity T can again bedenoted by n.

At 450, additional to the value τ₀, a set of n initial single-optionvalues {τ_(i)}₁ ^(n) of φ can be computed by the same procedure,excepting that the optimization is in accordance with a single optionprice in the group, and τ_(i) corresponds to the i-th option, ordered,for example, by ascending value of strike price.

As illustrated at 440, in the above-described procedure andsubsequently, if the set of option prices of selected maturity T is notcontemporaneous (within the set), one of the prices can be selected outof the corresponding paired set of contemporaneous underlying instrumentprices to be the current price S₀ for the purpose of furthercalculations. When implied option prices are calculated, the model canbe calibrated to determine an adjustment to the martingale distributionQ_(T) for each option for which the corresponding contemporaneousunderlying instrument price differs from S₀. Denoting the differingprice by S′, a counterpart adjustment can be made to Q_(T) for thepurpose of pricing the corresponding option. In one embodiment, forexample, the first moment of the distribution Q_(T) is set equal to theforward price corresponding to price S′ by shifting (e.g., adding to orsubtracting from) all prices over which the distribution is defined byan equal constant amount. The corresponding forward price can becomputed to be equal to the ratio of price S′ to the time T discountfactor (S′/DF_(T)), where the discount factor DF_(T) is as definedearlier.

d. Computing the Optimal Martingale Distribution O_(T)

In application to basket options (options written on a fixed weightportfolio of assets (such as stocks or currency exchange rates) pricedin a common currency), it may be that options on that particular baskethave not been traded hitherto, and for that or other reasons thoseparticular option prices are not available. In that case, the finalestimate of the optimal martingale distribution Q_(T) can be determinedby fixing a scalar value τ₀ of the distribution parameter function φ,based on an assessment of various factors, including, for example, thecorresponding initial optimal values {T₀} estimated from option pricesthat are available on similar baskets or on some or all of the componentassets of the particular basket.

At 451, an initial value of a Pick matrix is computed. The Pick matrix

an n-by-n matrix representing the options data, where again n denotesthe number of options in the group:

=(p _(K) _(j) _(K) _(k) )_(j,k,=1) ^(n),

and the subscripts {K_(j)} denote the strike prices. The off-diagonalelements p_(K) _(j) _(k) _(k) , j≠k, are defined by

$\begin{matrix}{p_{K_{j}K_{k}} = \frac{\tau_{j} - \tau_{k}}{K_{j} - K_{k}}} & {{Equation}\mspace{14mu} 19}\end{matrix}$

where the {τ_(i)}_(i=1) ^(n) are the set of n initial single-optionvalues of the distribution parameter function φ (and may be substitutedby subsequent values in subsequent iterations in an alternativeembodiment). The diagonal elements p_(K) _(j) _(K) _(j) represent upperlimits to limiting values of the off-diagonal elements (upper limits tothe gradient of the distribution parameter function φ with respect tothe strike price K). Such an upper bound (not necessarily optimal) canbe deduced from the maximum weight in the distribution Q (to bedetermined) that can be concentrated at any given point, consistent withthe given moments of the empirical distribution (consistent with theunderlying financial instrument data). The point is that a maximumexists for the weight, covering all possible distributions. Accordingly,the diagonal elements are taken to be

$\begin{matrix}{p_{K_{j}K_{j}} = {\sum\limits_{i}\frac{\rho \left( S_{i} \right)}{\left( {S_{i} - K_{j}} \right)^{2}}}} & {{Equation}\mspace{14mu} 20}\end{matrix}$

where the maximum weight ρ(S) is defined as earlier, and the sum istaken over all values {S_(i)} that are poles of the distributioncorresponding to φ=τ_(j). In the foregoing, the technical convenience isimposed (by slightly adjusting the distribution parameter function φ, ifnecessary) that the distribution to be determined does not have anypoint of increase at a strike price {K_(j)}; otherwise one would have towork explicitly with complex variables. Further, in a particularimplementation, the Pick matrix and its sub-matrices are required to bepositive and may be adjusted, if necessary, to ensure that condition.Also, in one embodiment, the upper bounds on the diagonal elements inthe Pick matrix can be tightened.

At 452, an initial value of a Potapov matrix is calculated. The Potapovmatrix is defined as follows. Let I denote the n-by-n identity matrix,let e denote the n-dimensional unit column vector:

${e = \begin{pmatrix}1 \\1 \\\vdots \\1\end{pmatrix}},$

and let ′ denote the transpose of a vector, so that e′=(1 1 . . . 1) isthe n-dimensional unit row vector. Let v_(φ) denote the n-dimensionalcolumn vector of the initial single-option values {τ_(i)}_(i=1) ^(n) ofthe distribution parameter function (which may be substituted bysubsequent values in subsequent iterations in an alternativeembodiment):

${v_{\varphi} = \begin{pmatrix}\tau_{1} \\\tau_{2} \\\vdots \\\tau_{n}\end{pmatrix}},$

so that v′₁₀₀ =(τ₁ τ₂ . . . τ_(n)). Let Z_(n) denote the n-by-n diagonalmatrix of strike prices in ascending order

$Z_{n} = {\begin{pmatrix}K_{1} & 0 & 0 & 0 \\0 & K_{2} & 0 & 0 \\0 & 0 & \ddots & 0 \\0 & 0 & 0 & K_{n}\end{pmatrix}.}$

The two-by-two Potapov matrix Θ(z) can be defined by the matrix equation

$\begin{matrix}{{\Theta (z)} = {\begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix} - {\begin{pmatrix}v_{\varphi}^{\prime} \\e^{\prime}\end{pmatrix}\left( {Z_{n} - z} \right)^{- 1}{{\mathbb{P}}_{n}^{- 1}\left( {e - v_{\varphi}} \right)}}}} & {{Equation}\mspace{14mu} 21}\end{matrix}$

This formula is the “infinite constant a→∞” version of Potapov's matrix.In an alternative embodiment, Θ^(a)(z), a finite constant a version ofPotapov's matrix can be used. A value of a is selected for computationalconvenience, subject to a>K_(n) (taking K_(n) to be the largest strikeprice), and the Potapov matrix is defined by

$\begin{matrix}{{\Theta^{a}(z)} = {\begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix} - {\left( {z - a} \right)\begin{pmatrix}v_{\varphi}^{\prime} \\e^{\prime}\end{pmatrix}\left( {Z_{n} - z} \right)^{- 1}{{\mathbb{P}}_{n}^{- 1}\left( {Z_{n} - {aI}} \right)}^{- 1}{\left( {e - v_{\varphi}} \right).}}}} & {{Equation}\mspace{14mu} 22}\end{matrix}$

Denoting the elements of the Potapov matrix by subscripts as follows

${{\Phi (z)} = \begin{pmatrix}{\Theta_{11}(z)} & {\Theta_{12}(z)} \\{\Theta_{21}(z)} & {\Theta_{22}(z)}\end{pmatrix}},$

the functional values of the distribution parameter function can bedetermined by values of the interpolation function U in accordance withthe fractional linear transformation given by the Potapov matrix:

$\begin{matrix}{{\varphi_{U}(z)} = \frac{{{\Theta_{11}(z)}U} + {\Theta_{12}(z)}}{{{\Theta_{21}(z)}U} + {\Theta_{22}(z)}}} & {{Equation}\mspace{14mu} 23}\end{matrix}$

which is referred to herein as the “interpolation formula”. A parallelformula is applied if Θ^(a)(z) is used in place of Θ(z). One feature ofembodiments of the disclosed technology is that even if the {s_(i)}₁₌₂^(∞) are assumed to remain constant (regarded as centered moments), theimplied option volatilities will vary with varying prices for theunderlying instrument (s_(i)) and for the options, because thetransformation matrices and the distribution parameter function vary.

Preferably, and in accordance with certain embodiments of the disclosedtechnology, the distribution parameter function is required tocorrespond to an n-canonical solution (φ is required to be a rationalfunction of exactly n-th degree (and real on the real axis)), where ndenotes the number of options in the selected group at maturity T. Byimplication, it follows that the interpolation function U necessarily isa scalar (a function equal to a fixed real number).

At 460, a set of test scalar values {U_(i)} of U is selected. The setcan include the value τ₀ (the initial value of φ) and can fall in arange around τ₀; the range of test values can be similar to that spannedby the maximum and minimum values in the set {τ_(i)}₁ ^(n).

At 462 and 464, from each of the set of scalar test values {U_(i)}, acorresponding test functional value φ_(U) of is determined, constitutingthe set {φ_(U) _(i) }, by means of the fractional linear transformationgiven by the Potapov matrix (the equation just given above).

Given a test functional value φ_(U) _(i) of φ, the implied price foreach option in the group is determined by means of the inverse Stieltjestransform sub-model (at 466) and the options pricing sub-model (at 468).(As before, the options pricing can include an adjustment fornon-contemporaneous option prices at 470.) Accordingly, the sets ofimplied option prices corresponding to each φ_(U) _(i) taken in turn arecomputed, together with each corresponding martingale distributionQ_(i).

At 469, the final estimate of the optimal martingale distribution Q_(T)is output. For example, the final estimate of the optimal martingaledistribution Q_(T) can be stored (e.g., on one or more non-transitoryreadable medium) and then displayed to a user using a suitable displaydevice.

e. Computing the Final Estimate of the Martingale Distribution Q_(T)

In one embodiment, the final estimate of the optimal martingaledistribution Q_(T) is taken to be the distribution Q_(i) that impliesoption prices that best approximate (or approximate within auser-selected or predetermined margin) the empirical data prices of theselected group of options, according to a predetermined or user-selectedcriterion (e.g. least squares).

Another embodiment is illustrated in FIGS. 5A-5B. In particular, FIGS.5A-5B show a flowchart 500 of an exemplary technique for determining afinal estimate of a martingale probability distribution by constructinga second value of the Pick matrix in accordance with an embodiment ofthe disclosed technology.

At 510, and using the probability distribution model as described above,test distributions {Q_(i)} at maturity T and corresponding to scalarvalues of the interpolation parameter function U are computed.

At 512, the implied option prices for options of maturity T are computedfor each Q_(i) (as described above at 466 and 468)

At 520 and 522, a subset of n or fewer distributions are selected fromthe set {Q_(i)} such that each Q_(i) implies an option price that bestapproximates (or approximates within a user-selected or predeterminedmargin) a single option price in the selected group, each price beingtaken in turn or in some other order.

At 530 and 532, the corresponding values of U_(i) are used to calculatecorresponding values φ_(i) of φ for the function valued at thecorresponding strike price of the option concerned.

From these values, and at 540, a second value of the Pick matrix isconstructed. As shown at 542, 544, 546, the method acts in thepreviously described method subsequent to the determination of theinitial value of the Pick matrix are followed, with the values {φ_(i)}substituted for the initial values {τ_(i)} of the distribution parameterfunction φ.

At 550, a final estimate of the distribution Q is determined. Forexample, in particular implementations, the final estimate is determinedaccording to the embodiment described previously, namely the Q_(i) thatimplies option prices that best approximate (or approximate within auser-selected or predetermined margin) the option prices of the selectedgroup taken together.

f. Computing Conditional Extremal Distributions Q_(H) and Q_(L)

Returning to FIG. 4, at 453, 454, 455, 456, the conditional extremaldistributions, conditional on the contemporaneous option price data, aregiven by extremal values of U, in analogy to the extremal values of thedistribution parameter function φ, namely U is set to be infinite orzero, corresponding respectively to

φ_(H)=Θ₁₁/Θ₂₁ Θ_(L)=₁₂/Θ₂₂   Equation 24

in order to generate measures that give respectively the conditionalhigh and low price bounds to options.

g. Further Considerations

Any of the method acts introduced above, from the selection of Tonwards, can be repeated for each group of options, corresponding toeach of the maturities {T}. As noted above, the final estimates ofdistributions {Q_(T)} correspond to a final set of scalar values {U_(T)}which may be smoothed as a function of maturity, so as to infer smoothedmartingale distributions that characterize the options market taken as awhole.

In certain embodiments of the disclosed technology, graphical displaysare generated of various quantities of interest, as functions of theunderlying instrument price S expressed in, for example, current valueterms. The various quantities of interest can additionally oralternatively be stored as data in one or more non-transitorycomputer-readable storage media. The quantities that are stored and/ordisplayed can include, for example, one or more of: the strategies{R*_(k)} (up to a predetermined or user-selected desired number); theassociated hedging coefficients delta, {Δ_(k)}, and gamma, {Γ_(k)}; and,for each maturity T, the martingale distribution Q_(T), the associatedoptimal upper and lower bounds to the distribution (Q_(F) and Q_(K),which are respectively the Friedrichs and Krein measures), and theconditional high and low bounds, Q_(H) and Q_(L), which are conditionalon the empirically given option prices. The displayed data can bedisplayed in a variety of formats (e.g., as part of a computer-displayedgraph or graphical user interface). The displayed data can also beprinted.

While the disclosed technology has been particularly shown and describedwith reference to preferred embodiments thereof, it will be understoodby those skilled in the art that various changes in form and details canbe made without departing from the spirit and scope of the invention. Inparticular, it will be understood that variants of the disclosedtechnology can include the following considerations: application tovarious asset classes, including stocks, interest rates, currencies,commodities, and credit; incorporation of dividend or interest payments;application to multi-asset options; application to American and mildlypath-dependent options and other forms of derivatives; use of put optionprices by means of put-call parity relations; adaptation to use of bothforward prices and discount factors in place of one of the foregoingtogether with current prices of the underlying financial instrument;adaptation to negative prices for financial instruments that do not havelimited liability; adaptation to use of non-equal time intervals betweenunderlying instrument price observations; and adaptation to use ofoption delta in place of strike price.

D. Generalized Embodiments of the Disclosed Technology

1. Generalized Methods for Generating Multiple Martingale ProbabilityDistributions

FIGS. 6A-6B show a flowchart 600 illustrating a generalized method forgenerating multiple martingale probability distributions for an optionor a group of options having a common maturity date. In particular,FIGS. 6A-6B illustrate a generalized version of the embodimentillustrated in FIG. 4. The exemplary method illustrated in FIGS. 6A-6Bshould not be construed as limiting, however, as any one or more of theillustrated acts can be performed alone or in various combinations andsubcombinations with other method acts.

In the exemplary method shown in FIGS. 6A-6B, historical price data isused only through the moments of a representative martingaledistribution at maturity T. The moments comprise the average of x, theaverage of x², the average x³, and so on. This is in contrast toalternative approaches in which historical price data is used implicitly(and in some instances explicitly) in the selection of a parametricfunctional form for a probability distribution. In effect, thefunctional form in such alternative approaches assumes more informationthan the data itself actually provides.

Furthermore, in the illustrated embodiment, martingale probabilitydistributions, computed in the form of “average components” followed byan inversion (e.g., integral and inverse integral transformations), arecomputed by algebraic manipulations. For instance, in certainimplementations of the illustrated embodiment, the computation of themartingale probability distributions is performed without using anystochastic or differential equation methods (e.g., the computation ofthe martingale probability distributions is performed using onlyalgebraic manipulations).

Further, in the illustrated embodiment, the input data used for variousmethod acts (e.g., the representative martingale distribution atmaturity T, and the short-run distribution) is input in the form of afinite set of discrete data, not a function. The moments of thosedistributions are then calculated and used in subsequent computations.

At 610, a representative martingale probability distribution of thefuture instrument price at maturity T is computed for an underlyingfinancial instrument. In certain embodiments, the computation isperformed using the historical price data for the underlying instrument(e.g., using only the historical price data for the underlyinginstrument). The historical price data 602 can be input (e.g., bufferedinto memory or otherwise loaded and prepared for further processing),for example, by accessing a suitable database (as described above inSection III.C.1 with respect to method act 210). Furthermore, thematurity date T can be input by a user as part of maturity data 604. Thematurity data 604 can comprise one or more maturity dates for individualmarket-traded options or groups of options. In particularimplementations, the maturity date T is provided by a user via asuitable graphical user interface. Further, the maturity date T can bebased on the available maturities of the market-traded options. (Itshould be noted that, in some instances, the representative distributionis termed “conditional” because it is conditional on the current marketprice of the underlying instrument. This usage should not be confusedwith “conditional on option prices”, which is used herein to refer tothe conditional extremal distributions.)

In the illustrated embodiment, the historical price data is used tocompute a short-run probability distribution, the moments of which arethen calculated. For purposes of this discussion, “moments” refer to oneor more of the average of x, the average of x², the average of x³, andso on. In particular embodiments (e.g., as described in SectionIII.C.3.a above), only the moments together with the current price ofthe underlying instrument are used to compute the representativedistribution (other historical price statistics are not used). Incertain embodiments, the computation of the representative distributionproceeds from the moments by calculating the corresponding tradingstrategies (e.g., in the form of orthogonal polynomials) and excessreturns.

In other embodiments, the empirical (as opposed to the martingale or“risk neutral”) distribution at maturity T is calculated from thehistorical price data. Then, the Esscher transform is used to computethe representative distribution (e.g., as described in Section III.C.3.aabove). The Esscher transform turns an empirical distribution into amartingale distribution, and can be computed in a straightforwardmanner.

In certain embodiments of the disclosed technology, the only way thatthe representative distribution is used in subsequent calculations isthrough its calculated moments. In technical terms, the momentsconstitute a sufficient statistic for the historical price data.Accordingly, and in some implementations, no parametric functional formsare used (or need to be used). In particular embodiments (and asdescribed in Section III.C.3.b), the representative distribution iscomputed in the form of a finite set of data points, without the use ofany parametric functional form, nor any stochastic or differentialequation methods. It should also be noted that, in the illustratedembodiment, the use of the moments used in process blocks 612-618 isseparate (and complementary) to the use of the moments of theunconstrained conditional price distribution shown at blocks 130-132.

At 612, extremal probability distributions are computed. In particularimplementations, the extremal probability distributions are thedistributions that give rise to upper and lower bounds on the price ofany option at maturity T for the underlying instrument. Thesedistributions are also referred to as the Friedrichs and Kreindistributions, respectively. In the illustrated embodiment, thedistributions are computed indirectly in the form of “averagecomponents” (or simply “averages”). The average components can be viewedas being analogous to frequency components in a frequency decompositionof a musical signal. In one exemplary embodiment, the average is theStieltjes transform of the distribution, and is an integraltransformation (e.g., as described in Section III.C.3.b above). Theaverage components for each distribution can be computed by a formulathat depends on both historical price data and a parameter that can bevaried. For instance, in the illustrated embodiment, the historicalprice data are used through the moments of the representative martingaledistribution (e.g., only through the moments of the representativemartingale distribution). In one particular exemplary implementation ofthe illustrated method (as described in Section III.C.3.b above), thehistorical price data is encapsulated in matrix form (e.g., theNevanlinna matrix), and the parameter that can be varied is termed thedistribution parameter function. In this implementation, the average isthe fractional linear transformation, determined by the Nevanlinnamatrix, of the distribution parameter function, which takes themathematical form of a compactified Nevanlinna class. The distributionparameter function has two extremal values, namely two fixed numbers.Substituting these numbers gives the average components of the extremaldistributions. The distributions themselves can be computed from theaverages by an inverse method (an inverse integral transformation). Forinstance, in one exemplary implementation, the inverse method is theinverse Stieltjes transform (also termed the Stieltjes-Perron inversionformula). By the inverse method, the extremal distributions arecomputed. In the particular implementation described above, thecomputations involving the historical price data and the inverse methodrequire only algebraic manipulations and, therefore, do not usestochastic or differential equation methods.

At 614, an initial estimate of the optimal martingale distribution(sometimes referred to as the “approximating martingale distribution”)at maturity T is computed. The approximating martingale distribution atmaturity T is the one that, when used to price the group ofmarket-traded options (the one or more options) of maturity T, gives thebest fit to contemporaneous market prices for options for whichcontemporaneous prices have been provided. By “best fit” is meant thevalue that best meets a predetermined or user-selected criterion (e.g.least squares, the fit that minimizes the sum of the squares of thedeviations between each computed price and its empirical datacounterpart, or other such criteria). Furthermore, in someimplementations, the best fit can be within some predetermined oruser-selected tolerance of the best possible value (e.g., within 1%, 2%,5%, or 10%). In some embodiments, during the computation of the initialestimate of the optimal distribution, option prices are not used tocompute the various candidate distributions, but only to select the bestfit distribution. Subsequent estimates (iterations) use the optionprices in the computation of the candidate distributions.

In one particular exemplary implementation, the computation proceeds asfollows. A set of scalar values (numbers) are selected for thedistribution parameter function. It should be noted that thedistribution parameter function can take not only scalar, but alsofunctional values. In other words, the distribution parameter function(e.g., distribution parameter function φ) can be not only a specificnumber, but also a specific function. At this stage of the initialestimate, however, the distribution parameter function is a scalar; whenoption prices are subsequently provided, the distribution parameterfunction is computed as a function (e.g., by means of the Potapov matrixand the interpolation parameter function). From the set of scalarvalues, a set of corresponding test distributions are computed. Inparticular, from each scalar value, a distribution is computed in theindirect form of “average components” (e.g., using the Stieltjestransform) by means of the incomplete markets formula (e.g., usingequation (14)). This formula uses historical prices in the form of theNevanlinna matrix (the historical prices can be said to be encapsulatedby the Nevanlinna matrix), which has been calculated at maturity T fromthe relevant moments. The “average components” form of each distributionis inverted to give the distribution by means of the inverse method(e.g., using the inverse Stieltjes transform). Then, for each testdistribution, the implied prices are computed for the providedmarket-traded options (the one or more options) of maturity T. Thecalculations are carried out by means of the options pricing sub-model.The options pricing sub-model is a pricing procedure that computes theprice of any option of the same maturity T given a martingaledistribution at maturity T. Exemplary options pricing sub-models thatcan be used include, for example, equation (17) above. Details of how tocompute the price according to the formula are known and need not be setforth here. In the particular implementation described above, thecomputations with the incomplete markets formula, the inverse method,and the options pricing sub-model require only algebraic manipulations,not stochastic or differential equation methods.

Each set of implied prices is compared to the provided set of marketprices for options of maturity T, and a best fit set is determined. Forexample, at 615 in the illustrated embodiment, the scalar distributionparameter function φ is varied to find the best fit of the impliedprices to the provided set of market prices. Thus, a best fit scalarvalue for the distribution parameter function is determined. If theprovided option prices at maturity T are not exactly contemporaneous(e.g., were prevailing at different times of the trading day) then thecalculated implied prices can be adjusted by known methods (e.g., asdescribed at Section III.C.3.c above). In one exemplary implementation,the best fit scalar value for the distribution parameter function isused subsequently merely as a guide for test values for theinterpolation parameter function. But the single-option best fit scalarvalues for the distribution parameter function, calculated at the sametime, are used directly to construct the Pick matrix (e.g., as describedat Sections III.C.3.d and III.C.3.e above).

At 616, two conditional extremal martingale distributions are computed.The two conditional extremal distributions are conditional on theprovided market prices of options at maturity T. In other words, theconditional extremal distributions incorporate the information from theoption prices (whereas the unconditional extremal distributions werebased only on historical price data for the underlying instrument). Ingeneral, the conditional extremal distributions provide tighter boundson upper and lower option prices than do the unconditional counterparts.At this stage of the computations, with the input of option prices, thedistribution parameter function is no longer constrained to be a scalar(a number), but may instead take functional values. This produces abetter distribution, as evidenced by a more accurate fit to market data.In one particular implementation (e.g., as described above at III.C.3.fand III.C.3.d), functional values of the distribution parameter functionare computed in two stages, both of which use the provided market optionprices (at maturity T). In the first stage, a mathematical operator iscalculated (e.g., the Pick matrix as described above in III.C.3.d). Inthe second stage, the distribution parameter function is given in termsof another parameter, the interpolation parameter function, by a formula(e.g., using the Potapov matrix, which depends on the Pick matrix, asdescribed above at III.C.3.d with respect to equation (23)). Theconditional extremal distributions are generated by the two extremalscalar values of the interpolation parameter function. The two extremalscalar values produce two corresponding functional values of thedistribution parameter function (e.g., using equation (23)). In turn,those two functional values are used in the incomplete markets formulato produce two indirect (“average components”) forms of thedistributions. In turn, the two indirect forms are used in the inversemethod (inverse Stieltjes transform) to produce two extremaldistributions. In the particular implementation described above, thecomputations require only algebraic manipulations, not stochastic ordifferential equation methods.

At 618, the optimal martingale distribution at maturity T is computed.The optimal distribution is computed by finding an optimal functionalvalue for the distribution parameter function. (In general, the use of afunctional value here improves the results from the initially estimatedscalar value.) Further, in the illustrated embodiment, the functionalvalue is computed using the option price data (for options of maturityT). In one particular implementation (e.g., as described above inSections III.C.3.d and II.C.3.e), functional values for the distributionparameter function are computed in two stages. In the first stage, amathematical operator (e.g., the Pick matrix, as described above inSection III.C.3.d) is calculated. In the second stage, the distributionparameter function is given in terms of another parameter, theinterpolation parameter function, by a formula (e.g., using the Potapovmatrix, which depends on the

Pick matrix, as described above in Section III.C.3.d). In thisimplementation, the mathematical operator (e.g., the Pick matrix) iscomputed from the options price data and expresses the gradient of thedistribution parameter function with respect to the strike price of theoptions (the one or more options) of maturity T. Accordingly, itrepresents a generalization of the volatility smile (the gradient ofvolatility against strike price) known to those skilled in the art. Byusing the Potapov matrix, functional values of the distributionparameter function are generated according to the values assigned to theinterpolation parameter function. In certain desirable implementations(e.g., as described above in Section III.C.3.d), the interpolationparameter function is taken to be a scalar value. In suchimplementations, any number of candidate probability distributions (anddesirably all candidate probability distributions) can be generated by asingle scalar parameter (the interpolation parameter function). Tocompute the candidate probability distributions, test scalar values ofthe interpolation parameter function are selected (e.g., two or moretest scalar values). The test scalar values can be selected by using thepreviously fitted scalar values (aggregate and single option) of thedistribution parameter function or by some other means (e.g., randomly,predetermined, or user input). From each of the set of scalar testvalues, a corresponding test functional value of the distributionparameter function is determined. From each test value of thedistribution parameter function, again functional values, an indirect(“average components”) form of the corresponding test distribution iscomputed by means of the incomplete markets formula. From the indirectform, the distribution is computed by using the inverse method (e.g., aninverse Stieltjes transform). Then, for each test distribution, theimplied prices are calculated of the provided market-traded options (theone or more options) of maturity T. The calculations are carried out bymeans of the options pricing sub-model. In the particular implementationdescribed above, the computations with the certain formula, the Pick andPotapov matrices, the incomplete markets formula, the inverse method,and the options pricing sub-model require only algebraic manipulations,not stochastic or differential equation methods.

Each set of implied prices is compared to the provided set of marketprices for options of maturity T, and a best fit set is determined. Forexample, at 619 in the illustrated embodiment, the interpolationparameter function U is varied to find the best fit of the impliedprices to the provided set of market prices through a functional valuefor the distribution parameter function φ. Thus, a best fit scalar valuefor the interpolation parameter function (and a functional value for thedistribution parameter function) is determined.

At 620, the martingale probability distributions at each selectedmaturity T are stored or stored and displayed (e.g., as described aboveat Section III.C.3.g). In the illustrated embodiment, the martingaleprobability distributions stored and/or displayed comprise the optimaldistribution, the two extremal distributions, and the two conditionalextremal distributions. In particular implementations, the martingaleprobability distributions are stored and displayed as distributions ofthe underlying financial instrument price, expressed in current valueterms (as a price prevailing at date T) (e.g., as described above). Thedisplayed data can be displayed in a variety of formats (e.g. as part ofa computer-displayed graph or graphic user interface) and on a varietyof different display devices (e.g., computer monitors, touchscreendisplays, tablet displays, and the like). The displayed data can also beprinted.

2. Generalized Methods for Generating Multiple Trading Strategies,Associated Hedging Coefficients, and Expected Excess Returns

FIG. 7 is a flowchart 700 illustrating an exemplary method forgenerating one or more of trading strategies, hedging coefficients, orexpected excess returns for an underlying financial instrument. Inparticular, FIG. 7 illustrates a generalized version of the embodimentdescribed above with respect to FIG. 2. The exemplary method illustratedin FIG. 7 should not be construed as limiting, however, as any one ormore of the illustrated acts can be performed alone or in variouscombinations and subcombinations with other method acts.

The method illustrated by flowchart 700 uses historical price data 702(a time series) for the underlying financial instrument, including thecurrent market price, as input. The historical price data can be input(e.g., buffered into memory or otherwise loaded and prepared for furtherprocessing), for example, by accessing a suitable database (as describedabove in Section III.C.1 with respect to method act 210). The methodillustrated by flowchart 700 also uses a returns frequency 704,equivalently a time interval, as input. The returns frequency (or timeinterval) can be received as part of user data generated by a userselecting the returns frequency through a graphical user interface orother interface with a user. In particular implementations, the minimumtime interval is the time between data observations, other intervalsbeing integral multiples of that minimum. The overall length of the timeseries can also be received as part of user data generated by a userselecting the length through a graphical user interface or otherinterface. Alternatively, any one or more of the returns frequency orthe length of the time series can be predetermined.

At 710, from one or more pairs of price observations separated by theselected time interval (e.g., from every pair of price observations),the short-run return is calculated. For example, in one particularimplementation, if there are 1,000 consecutive daily price observations,and the selected time interval is daily, the number of returnscalculated is 999; if the interval is 2 days, the number is 998. The setof returns so obtained constitutes the returns density (a probabilitydensity). In one exemplary implementation, an initial short-runprobability distribution, the unconstrained conditional pricedistribution, is computed as follows. (This distribution, like allothers, is a distribution over the underlying financial instrumentprice.) The returns density acting over the selected time interval onthe current market price of the underlying instrument generates theinitial short-run probability distribution (e.g., as described above inSection III.C.1 and III.C.2). The distribution is termed “conditional”because it is conditional on the current market price of the underlyingfinancial instrument. It is termed unconstrained because there is atthis stage no constraint on returns observations (e.g. as to whetherthey arise from prices close to the current market price of theunderlying instrument). The constrained short-run probabilitydistribution, meaning the constrained distribution which will be used insubsequent calculations, can be computed using a variety of methods. Forexample, in one embodiment, the Whittle estimator is used (e.g., asdescribed above at Section III.C.2). In this embodiment, a “bandwidth”is selected that determines which observations from the historical pricedata are to be retained (“the constrained data”), dependent on whetherthe returns were generated from a price close to the current marketprice of the underlying instrument. The unconstrained distribution (theinitial short-run distribution) is modified by the technique oforthogonal series. In a particular implementation, the calculation usesorthogonal polynomials whose value is determined solely by theconstrained data. Other embodiments can also be used (e.g., as describedabove with respect to method act 220).

In the illustrated embodiment, the moments of the short-run distributionare computed and used in subsequent computations. In particularimplementations of the illustrated embodiment, only the moments areused, not any other statistics of the distribution, nor any parametricfunctional form for it. For instance, the short-run distribution can becomputed in the form of a finite set of data points, without the use ofany parametric functional form, nor any stochastic or differentialequation methods. The number of moments computed can be predetermined oruser selected.

At 712, the trading strategies are computed from the moments. Thetrading strategies form a long sequence (e.g., an endless sequence) andso only a pre-determined or user-selected number are computed. Ingeneral, the maximum number of trading strategies is limited by thenumber of moments calculated. In one exemplary implementation, thetrading strategies are calculated recursively as polynomials, orthogonalto each other with respect to the short-run distribution (e.g., asdescribed above in Section III.C.1 with respect to method act 242). Inother words, given a starting strategy, each successive strategy isuniquely determined by the condition that it be orthogonal to previouslycomputed strategies (e.g., all previously computed strateges) withrespect to the short-run distribution. (Orthogonal means that theexpected value, under the distribution, of the product of any twostrategies equals zero.) The starting strategy (the zeroth order one) istypically taken to be the function 1 (referring to investment in thecash asset (which has a value of 1 per unit investment)). Theindependent variable in the polynomials is the underlying instrumentprice. In other words, the trading strategies are expressed aspolynomial functions of the underlying instrument price. The tradingstrategies are thus calculated as explicit functions of the underlyinginstrument market price. The function indicates the value of thestrategy (investment position) for any given value of the instrumentprice. That is, the value of the trading strategy is indicated by theexplicit function of the instrument price. Because the tradingstrategies are constructed to be orthogonal, they are uncorrelated(which is a desirable features since traders want to be able to separateout their risks). The hedging coefficients are computed in explicitfunctional form as functions of the underlying instrument price (e.g.,as explained above in Section III.C.1 with respect to method act 250).At least two coefficients can be calculated for each trading strategy,the delta coefficient and the gamma coefficient, which are coefficientscommonly used by investors in making investment decisions. The expectedexcess return for each trading strategy can be calculated (as a fixednumber) (e.g., as described above in Section III.C.1 with respect tomethod act 280). The expected excess return is the excess return (percent per annum) expected over and above the return to the cash strategy(an investment in cash (a money market instrument)). Because the tradingstrategies, associated (corresponding) hedging coefficients, andexpected excess returns are computed purely from the moments of theshort-run distribution, they can be determined regardless of whether ornot options are traded on the financial instrument. That is, the tradingstrategies do not require the input of any option price data.

In an alternative embodiment, the trading strategies are calculated inthe same manner as the exemplary embodiment described above, exceptingthat they are calculated with respect to the representative martingaledistribution at a selected maturity T, not the short-run distribution.The computations of hedging coefficients and expected excess returnsalso proceeds in the same manner, excepting that they are based on therepresentative martingale distribution.

At 714, any one or more of the trading strategies, associated(corresponding) hedging coefficients, or excess returns are stored orstored and displayed. For example, the trading strategies and hedgingcoefficients can be stored and displayed as functions of the underlyingfinancial instrument price, expressed in current value terms (as a priceprevailing at date T) (e.g., as described above in connection with FIG.3). The displayed data can be displayed in a variety of formats (e.g. aspart of a computer-displayed graph or graphic user interface) and on avariety of different display devices (e.g., computer monitors,touchscreen displays, tablet displays, and the like). The displayed datacan also be printed.

IV. Exemplary Computing Environments for Implementing Embodiments of theDisclosed Technology

Any of the disclosed methods can be implemented as computer-executableinstructions stored on one or more computer-readable media (e.g.,non-transitory computer-readable media, such as one or more opticalmedia discs, volatile memory components (such as DRAM or SRAM), ornonvolatile memory components (such as hard drives)) and executed on acomputer (e.g., any commercially available computer, including smartphones, tablet computers, netbooks, or other devices that includecomputing hardware). Any of the computer-executable instructions forimplementing the disclosed techniques as well as any data created andused during implementation of the disclosed embodiments (e.g., the inputdata, or any one or more of the generated intermediate probabilitydistributions or their moments, associated “a and b coefficients” orexpectations generator, trading strategies or associated expected excessreturns or hedging coefficients, Nevanlinna, Pick or Potapov matrices,extremal or optimal martingale distributions, distribution parameter orinterpolation functions, or other elements) can be stored on one or morecomputer-readable media (e.g., non-transitory computer-readable media).The computer-executable instructions can be part of, for example, adedicated software application or a software application that isaccessed or downloaded via a web browser or other software application(such as a remote computing application). Such software can be executed,for example, on a single local computer (e.g., any suitable commerciallyavailable computer) or in a network environment (e.g., via the Internet,a wide-area network, a local-area network, a client-server network (suchas a cloud computing network), or other such network) using one or morenetwork computers.

For clarity, only certain selected aspects of the software-basedimplementations are described. Other details that are well known in theart are omitted. For example, it should be understood that the disclosedtechnology is not limited to any specific computer language or program.For instance, the disclosed technology can be implemented by softwarewritten in C++, Java, Perl, JavaScript, Adobe Flash, or any othersuitable programming language. Likewise, the disclosed technology is notlimited to any particular computer or type of hardware. Certain detailsof suitable computers and hardware are well known and need not be setforth in detail in this disclosure.

Furthermore, any of the software-based embodiments (comprising, forexample, computer-executable instructions for causing a computer toperform any of the disclosed methods) can be uploaded, downloaded, orremotely accessed through a suitable communication means. Such suitablecommunication means include, for example, the Internet, the World WideWeb, an intranet, software applications, cable (including fiber opticcable), magnetic communications, electromagnetic communications(including RF, microwave, and infrared communications), electroniccommunications, or other such communication means.

The disclosed methods can also be implemented by specialized computinghardware that is configured to perform any of the disclosed methods. Forexample, the disclosed methods can be implemented by an integratedcircuit (e.g., an application specific integrated circuit (“ASIC”) orprogrammable logic device (“PLD”), such as a field programmable gatearray (“FPGA”)). The integrated circuit can be embedded in or directlycoupled to an electrical device having a suitable display configured todisplay.

FIG. 8 illustrates a generalized example of a suitable computingenvironment 800 in which several of the described embodiments can beimplemented. The computing environment 800 is not intended to suggestany limitation as to the scope of use or functionality of the disclosedtechnology, as the techniques and tools described herein can beimplemented in diverse general-purpose or special-purpose environmentsthat have computing hardware.

With reference to FIG. 8, the computing environment 800 includes atleast one processing unit 810 and memory 820. In FIG. 8, this most basicconfiguration 830 is included within a dashed line. The processing unit810 executes computer-executable instructions. In a multi-processingsystem, multiple processing units execute computer-executableinstructions to increase processing power. The memory 820 may bevolatile memory (e.g., registers, cache, RAM), non-volatile memory(e.g., ROM, EEPROM, flash memory), or some combination of the two. Thememory 820 stores software 880 implementing one or more of the describedtechniques for operating or using the disclosed technology. For example,the memory 820 can store software 880 for implementing any of thedisclosed techniques described herein and their accompanying userinterfaces.

The computing environment can have additional features. For example, thecomputing environment 800 includes storage 840, one or more inputdevices 850, one or more output devices 860, and one or morecommunication connections 870. An interconnection mechanism (not shown)such as a bus, controller, or network interconnects the components ofthe computing environment 800. Typically, operating system software (notshown) provides an operating environment for other software executing inthe computing environment 800, and coordinates activities of thecomponents of the computing environment 800.

The storage 840 can be removable or non-removable, and includes magneticdisks, magnetic tapes or cassettes, CD-ROMs, DVDs, or any other tangiblenon-transitory non-volatile memory or storage medium which can be usedto store information and which can be accessed within the computingenvironment 800. The storage 840 can also store instructions for thesoftware 880 implementing any of the described techniques, systems, orenvironments.

The input device(s) 850 can be a touch input device such as a keyboard,touchscreen, mouse, pen, trackball, a voice input device, a scanningdevice, or another device that provides input to the computingenvironment 800. The output device(s) 860 can be a display device (e.g.,a computer monitor, smartphone display, tablet display, netbook display,or touchscreen), printer, speaker, CD-writer, or another device thatprovides output from the computing environment 800.

The communication connection(s) 870 enable communication over acommunication medium to another computing entity. The communicationmedium conveys information such as computer-executable instructions,resource allocation messages or data, or other data in a modulated datasignal. A modulated data signal is a signal that has one or more of itscharacteristics set or changed in such a manner as to encode informationin the signal. By way of example, and not limitation, communicationmedia include wired or wireless techniques implemented with anelectrical, optical, RF, infrared, acoustic, or other carrier.

As noted, the various methods can be described in the general context ofcomputer-readable instructions stored on one or more computer-readablemedia. Computer-readable media are any available media that can beaccessed within or by a computing environment. By way of example, andnot limitation, with the computing environment 800, computer-readablemedia include tangible non-transitory computer-readable media such asmemory 820 and storage 840.

The various methods disclosed herein can also be described in thegeneral context of computer-executable instructions, such as thoseincluded in program modules, being executed in a computing environmentby a processor. Generally, program modules include routines, programs,libraries, objects, classes, components, data structures, and so on thatperform particular tasks or implement particular abstract data types.The functionality of the program modules may be combined or splitbetween program modules as desired in various embodiments.Computer-executable instructions for program modules may be executedwithin a local or distributed computing environment.

An example of a possible network topology (e.g., a client-servernetwork) for implementing a system according to the disclosed technologyis depicted in FIG. 9. Networked computing devices 920, 922, 930, 932can be, for example, computers running browser or other software thatcommunicates with one or more central computers 910 via network 912. Thecomputing devices 920, 922, 930, 932 and the central computer 910 canhave computer architectures as shown in FIG. 8 and discussed above. Thecomputing devices 920, 922, 930, 932 are not limited to traditionalpersonal computers but can comprise other computing hardware configuredto connect to and communicate with a network 912 (e.g., smart phones orother mobile computing devices, servers, dedicated devices, and thelike).

In the illustrated embodiment, the computing devices 920, 922, 930, 932are configured to communicate with one or more central computers 910(e.g., using a cloud network or other client-server network). In certainimplementations, the central computers 910 computes one or more of theintermediate or final values associated with the disclosed embodiments(e.g., any one or more of the generated intermediate probabilitydistributions or their moments, associated “a and b coefficients” orexpectations generator, trading strategies or associated expected excessreturns or hedging coefficients, Nevanlinna, Pick or Potapov matrices,extremal or optimal martingale distributions, distribution parameter orinterpolation functions, or other elements) and transmits theintermediate or final values to any of the computing devices 920, 922(e.g., as data to be displayed on a graphic user interface or web pageat the computing devices 920, 922). For example, the computing devices920, 922 can transmit a request for the data to the central computer 910over the network 912. In order to compute the intermediate or finalvalues, the one or more central computers 910 can access data fromcomputing devices 930, 932, which can store various types of marketdata. For example, the computing devices 930, 932 may store historicalprice information that is used in connection with the disclosedembodiments. Alternatively, the one or more central computers 910 maythemselves store the historical price information used in connectionwith the disclosed embodiments.

Another example of a possible network topology for implementing a systemaccording to the disclosed technology is depicted in FIG. 10. Networkedcomputing devices 1020, 1022, 1030, 1032 can be, for example, computersrunning browser or other software connected to a network 1012. As above,the computing devices 1020, 1022, 1030, 1032 can have computerarchitectures as shown in FIG. 8 and discussed above. The computingdevices 1020, 1022, 1030, 1032 are not limited to traditional personalcomputers but can comprise other computing hardware configured toconnect to and communicate with a network 1012 (e.g., smart phones orother mobile computing devices, servers, dedicated devices, and thelike).

In the illustrated embodiment, the computing devices 1020, 1022 areconfigured to communicate directly with computing devices 1030, 1032 viathe network 1012. In the illustrated embodiment, the computing devices1020, 1022 are configured to locally compute one or more of theintermediate or final values associated with the disclosed embodiments(e.g., any one or more of the generated intermediate probabilitydistributions or their moments, associated “a and b coefficients” orexpectations generator, trading strategies or associated expected excessreturns or hedging coefficients, Nevanlinna, Pick or Potapov matrices,extremal or optimal martingale distributions, distribution parameter orinterpolation functions, or other elements) using data obtained from thecomputing devices 1030, 1032 via the network 1012 (e.g., historicalprice information). Any of the intermediate or final values can bestored or displayed on any of the computing devices 1020, 1022 (e.g.,displayed as data on a graphic user interface or web page at thecomputing devices 1020, 1022).

In the illustrated embodiments, the illustrated networks 912, 1012 canbe implemented as a Local Area Network (“LAN”) using wired networking(e.g., the Ethernet IEEE standard 802.3 or other appropriate standard)or wireless networking (e.g. one of the IEEE standards 802.11a, 802.11b,802.11g, or 802.11n or other appropriate standard). Alternatively, andmost likely, at least part of the networks 912, 1012 can be the Internetor a similar public network and operate using an appropriate protocol(e.g., the HTTP protocol).

V. Additional Embodiments

It should be understood that the embodiments introduced above arerepresentative embodiments that are not intended to be limiting in anyway. Instead, any one or more aspects of the disclosed technology can beapplied in other embodiments, such as the embodiments described below.

In one exemplary embodiment, a computer-implemented method comprisesproducing complete option pricing in an incomplete market by explicitlymodelling the nature of the incomplete market so as to determine a rangeof all possible prices for an option on an underlying financialinstrument. The underlying financial instrument can be one of a singleasset, a single asset class, or a fixed-weight portfolio. In certainimplementations, the method is performed without using parametricfunctional forms for the underlying instrument's price process or riskpremium. Further, in some embodiments,the method is performed withoutusing differential or stochastic equations, or their associatednumerical solution techniques. In certain embodiments, the methodfurther comprises determining a set of martingale probabilitydistributions corresponding to the possible prices and causing one ormore of the distributions to be displayed on a display device.

In another exemplary embodiment, a computer-implemented method comprisesreceiving historical prices for an underlying instrument, and derivingone or more risk factors from a probability distribution of short-runfuture prices of the underlying instrument, wherein the probabilitydistribution is based on historical price data for the underlyinginstrument. One or more of the risk factors can be determined by themoments of the probability distribution of short-run future prices. Incertain embodiments, the method is performed without using parametricfunctional forms for the underlying instrument's price process or riskpremium. In some embodiments, the probability distribution of short-runfuture prices is determined using the technique of orthogonal series. Incertain embodiments, the risk factors are obtained as explicitanalytical functions of the underlying instrument price. In particularembodiments, the method is performed without using differential orstochastic equations, or their associated numerical solution techniques.In certain embodiments, the method is performed without usingcontemporaneous price data for options on the underlying instrument.

In another exemplary embodiment, a computer-implemented method comprisesdetermining an expectations generator that generates expected presentvalue pricing from the probability distribution of short-run futureprices. In certain embodiments, the expectations generator is determinedby the moments of the probability distribution of short-run futureprices. In some embodiments, the method is performed without usingparametric functional forms for the underlying instrument's priceprocess or risk premium. In certain embodiments, the method is performedwithout using differential or stochastic equations, or their associatednumerical solution techniques. In some embodiments, the method isperformed without using contemporaneous price data for options on theunderlying instrument.

In a further exemplary embodiment, a computer-implemented methodcomprises determining one or more trading strategies, associated hedgingcoefficients, associated excess returns, or any combination of tradingstrategies, associated hedging coefficients, and associated excessreturns based on one or more of the risk factors and the expectationsgenerator; and causing one or more of the trading strategies, theassociated hedging coefficients, the associated excess returns, or anycombination of the trading strategies, the associated hedgingcoefficients, and the associated excess returns to be displayed on adisplay device. In certain embodiments, one or more of the risk factorsare used as uncorrelated trading strategies in the underlyinginstrument. In some embodiments, the associated hedging coefficientsdelta and gamma are obtained as explicit analytical functions of theunderlying instrument price. In certain embodiments, the associatedexcess returns are obtained as the eigenvalues of the expectationsgenerator. In some embodiments, the method is performed without usingparametric functional forms for the underlying instrument's priceprocess or risk premium. In some embodiments, the method is performedwithout using differential or stochastic equations, or their associatednumerical solution techniques. In certain embodiments, the method isperformed without using contemporaneous price data for options on theunderlying instrument.

In another exemplary embodiment, a computer-implemented method comprisesgenerating a representative martingale probability distributionpertaining to a future time, corresponding to an option expiration date,by combining the risk factors and risk premia. In some embodiments, thetime evolution of each risk factor is determined by the correspondingrisk premium. In certain embodiments, the initial combination of riskfactors is determined uniquely. In some embodiments, the method isperformed without using parametric functional forms for an underlyinginstrument's price process or risk premium. In certain embodiments, themethod is performed without using differential or stochastic equations,or their associated numerical solution techniques. In some embodiments,the method is performed without using contemporaneous price data foroptions on the underlying instrument. In certain embodiments, thecombination of risk factors and risk premia is replaced alternatively bya given empirical probability distribution pertaining to the futuretime, which distribution may be derived from the set of historicalprices for the underlying instrument, and the Esscher transform of thegiven distribution (the exponentially tilted distribution) is used as arepresentative of all the distributions bearing the same moments as therepresentative martingale probability distribution.

In a further exemplary embodiment, a computer-implemented methodcomprises determining one or more martingale distributions pertaining toa future time that can be used to price an option with correspondingexpiration date on an underlying instrument by using a single function,wherein the relationship of each of the one or more distributions to thesingle function is based on historical price data for the underlyinginstrument. In certain embodiments, the relationship of each of the oneor more distributions to the single function is based on the moments ofthe representative martingale distribution. In some embodiments, therelationship of each of the one or more distributions to the singlefunction is that of an inverse integral transform of a fractional lineartransformation of the single function. In certain embodiments, thesingle function belongs to the compactified Nevanlinna class and has twoextremal values that determine extremal distributions and associatedupper and lower bounds to a price of an option on the underlyinginstrument. In some embodiments, the method is performed without usingparametric functional forms for the underlying instrument's priceprocess or risk premium. In certain embodiments, the method is performedwithout using differential or stochastic equations, or their associatednumerical solution techniques. In some embodiments, the method isperformed without using contemporaneous price data for options on theunderlying instrument. In certain embodiments, wherein the methodfurther comprises pricing a basket option using the single functionwithout any historical price data for the basket option, wherein a valueis assigned to the single function. In some embodiments, the methodfurther comprises causing one or more of the future martingaledistributions, which may include the extremal distributions, to bedisplayed on a display device.

In another exemplary embodiment, a computer-implemented method comprisesreceiving contemporaneous prices for one or more of a plurality ofoptions with a single expiration date on an underlying instrument; anddetermining one or more martingale distributions pertaining to theexpiration date that can be used to price an option with the expirationdate on the underlying instrument by using a single scalar parameter,wherein the relationship of the single function corresponding to each ofthe one or more martingale distributions to the single scalar parameteris based on both historical price data for the underlying instrument andcontemporaneous price data for one or more of the plurality of options.In certain embodiments, the relationship of the single functioncorresponding to each of the one or more martingale distributions to thesingle scalar parameter is that of a fractional linear transformation ofthe single scalar parameter. In some embodiments, the scalar parameterhas two extremal values that determine conditional upper and lowerbounds to the price of any option on the underlying instrument,conditional on the contemporaneous option prices. In certainembodiments, the method is performed without using parametric functionalforms for the underlying instrument's price process or risk premium. Insome embodiments, the method is performed without using differential orstochastic equations, or their associated numerical solution techniques.In certain embodiments, the method further comprises causing one or moreof the future martingale distributions, which may include theconditional extremal distributions, to be displayed on a display device.In some embodiments, the method further comprises determining an optimalmartingale distribution from among the distributions, wherein the singlescalar parameter is set equal to a value that implies option prices forone or more of the plurality of options that approximate thecorresponding contemporaneous prices. In certain embodiments, the methodfurther comprises causing the optimal martingale distribution to bedisplayed on a display device.

In a further exemplary embodiment, a method for creating tradingstrategies in an underlying financial instrument and derivingprobability distributions of the price of the underlying financialinstrument at future times comprises: (a) providing a probabilitydistribution model having a distribution parameter function,constituting at least one distribution parameter; (b) providing a set ofhistorical prices for the underlying financial instrument; (c) providinga set of contemporaneous prices for a plurality of options on theunderlying financial instrument, the options being associated to a setof strike prices and expiration dates, and accompanied by a paired setof prices for the underlying financial instrument, each pair of pricesbeing contemporaneous; (d) analyzing the set of historical prices todetermine a probability distribution of short-run future prices of theunderlying financial instrument; (e) determining a set of tradingstrategies in the underlying financial instrument in accordance with anexpectations generator matrix; (f) determining a set of hedgingcoefficients and excess returns, corresponding to the tradingstrategies; (g) generating a probability distribution of futurefinancial instrument prices pertaining to a selected time correspondingto one of the expiration dates of the plurality of options; (h)determining two extremal probability distributions of future financialinstrument prices in accordance with an incomplete markets formula; (i)analyzing a subset, defined by the selected time, of the contemporaneousprices for the plurality of options to determine a distributionparameter function value in accordance with an interpolation formula;(j) determining two conditional extremal probability distributions offuture financial instrument prices, conditional on the set ofcontemporaneous prices; (k) defining an optimal probability distributionthat determines option prices approximating the subset ofcontemporaneous prices using the probability distribution model and thedistribution parameter function value; (l) displaying graphically on adisplay device the optimal probability distribution, and the extremaland conditional extremal probability distributions; (m) displayinggraphically on the display device the trading strategies and theassociated hedging coefficients and excess returns, wherein thedetermined trading strategies, hedging coefficients and excess returnsare used in an investment portfolio to provide excess returns and in anoptions portfolio to provide optimal hedging, the determined extremaland conditional extremal probability distributions are used in an optionpricing model to provide upper and lower bounds to the price of aspecified option on the underlying financial instrument, and the definedoptimal probability distribution is used in an option pricing model toprovide a price of a specified option on the underlying financialinstrument.

VI. Concluding Remarks

Having illustrated and described the principles of the disclosedtechnology, it will be apparent to those skilled in the art that thedisclosed embodiments can be modified in arrangement and detail withoutdeparting from such principles. In view of the many possible embodimentsto which the principles of the disclosed technologies can be applied, itshould be recognized that the illustrated embodiments are only preferredexamples of the technologies and should not be taken as limiting thescope of the invention. Rather, the scope of the invention is defined bythe following claims and their equivalents. I therefore claim all thatcomes within the scope and spirit of these claims.

1. A computer-implemented method, comprising: with a computing device,receiving historical price data, the historical price data indicatingone or more historical prices for a financial instrument; receivingmaturity data, the maturity data indicating one or more expirationdates, each of the one or more expiration dates being an expiration datefor one or more options on the financial instrument; selecting anexpiration date from the one or more expiration dates; computing a setof two extremal martingale probability distributions based at least inpart on the historical price data and the selected expiration date, eachof the two extremal martingale probability distributions indicatingprobabilities of possible prices for the financial instrument at theselected expiration date, wherein each of the two extremal martingaleprobability distributions is determined by a fractional lineartransformation of a distribution parameter function; and storing the setof two extremal martingale probability distributions.
 2. The method ofclaim 1, wherein the computing the set of two extremal martingaleprobability distributions comprises setting the value of thedistribution parameter function to each of two pre-determined fixedscalar values.
 3. The method of claim 1, wherein the computing the setof two extremal martingale probability distributions comprises:computing a representative martingale probability distribution using thehistorical price data and the selected expiration date, therepresentative martingale probability distribution indicatingprobabilities of possible prices for the financial instrument at theselected expiration date; computing one or more mathematical moments ofthe representative martingale probability distribution; computing thefractional linear transformation of the distribution parameter function,the fractional linear transformation being based at least in part on theone or more mathematical moments; assigning values to the distributionparameter function; and computing the set of two extremal martingaleprobability distributions based at least in part on the fractionallinear transformation and the values assigned to the distributionparameter function.
 4. The method of claim 3, wherein the mathematicalmoments of the representative martingale probability distribution arecomputed by an Esscher transform of an empirical probabilitydistribution.
 5. The method of claim 3, wherein the set of two extremalmartingale probability distributions is based on the one or moremathematical moments of the representative martingale probabilitydistribution, and not on any other use of the historical price data. 6.The method of claim 1, wherein the computing the set of two extremalmartingale probability distributions is performed using only algebraicmanipulations.
 7. The method of claim 1, wherein the computing the setof two extremal martingale probability distributions is performedwithout using either stochastic computations or differential equationcomputations.
 8. The method of claim 1, wherein the method furthercomprises receiving a forward price or discount factor for the financialinstrument at the selected expiration date, and wherein the computingthe set of two extremal martingale probability distributions is furtherbased at least in part on the forward price or the discount factor. 9.The method of claim 1, wherein the set of two extremal martingaleprobability distributions comprises: a first martingale probabilitydistribution indicating lower bounds of possible prices for an option onthe financial instrument at the selected expiration date, the lowerbounds being independent of current option prices for the financialinstrument; and a second martingale probability distribution indicatingupper bounds of possible prices for an option on the financialinstrument at the selected expiration date, the upper bounds beingindependent of current option prices for the financial instrument. 10.One or more non-transitory computer-readable media storingcomputer-executable instructions which when executed by a computer causethe computer to perform the method of claim
 1. 11. A method, comprising:with a computing device, receiving historical price data, the historicalprice data indicating one or more historical prices for a financialinstrument; receiving maturity data, the maturity data indicating one ormore expiration dates, each of the one or more expiration dates being anexpiration date for one or more options on the financial instrument;selecting an expiration date from the one or more expiration dates;receiving option prices, the option prices indicating a respectivemarket price for each of one or more available options on the financialinstrument, each of the option prices being contemporaneous orsubstantially contemporaneous with one another, and each of theavailable options having an expiration date coinciding with the selectedexpiration date; computing a set of two conditional extremal martingaleprobability distributions based at least in part on the historical pricedata, the selected expiration date, and the option prices, each of thetwo conditional extremal martingale probability distributions indicatingconditional probabilities of possible prices for the financialinstrument at the selected expiration date, the conditionalprobabilities being conditional on the option prices, wherein each ofthe two conditional extremal martingale probability distributions isdetermined by a fractional linear transformation of a distributionparameter function; and storing the set of two conditional extremalmartingale probability distributions.
 12. The method of claim 11,wherein the computing the set of two conditional extremal martingaleprobability distributions comprises: computing initial estimates ofapproximating martingale probability distributions based on test scalarvalues of the distribution parameter function; and setting the value ofan interpolation parameter function to each of two pre-determined fixedscalar values.
 13. The method of claim 11, wherein the computing the setof two conditional extremal martingale probability distributionscomprises: computing a representative martingale probabilitydistribution using the historical price data, the representativemartingale probability distribution indicating probabilities of possibleprices for the financial instrument at the selected expiration date;computing one or more mathematical moments of the representativemartingale probability distribution; computing the fractional lineartransformation of the distribution parameter function, the fractionallinear transformation being based at least in part on the one or moremathematical moments; assigning values to the distribution parameterfunction; assigning values to an interpolation parameter function; andcomputing the set of two conditional extremal martingale probabilitydistributions based at least in part on the fractional lineartransformation, the values assigned to the distribution parameterfunction, and the values assigned to the interpolation parameterfunction.
 14. The method of claim 13, wherein the one or moremathematical moments of the representative martingale probabilitydistribution are computed by an Esscher transform of an empiricalprobability distribution.
 15. The method of claim 13, wherein the set oftwo conditional extremal martingale probability distributions is basedon the one or more mathematical moments of the representative martingaleprobability distribution, and not on any other use of the historicalprice data.
 16. The method of claim 11, wherein the computing the set oftwo conditional extremal martingale probability distributions isperformed using only algebraic manipulations.
 17. The method of claim11, wherein the computing the set of two conditional extremal martingaleprobability distributions is performed without using either stochasticcomputations or differential equation computations.
 18. The method ofclaim 11, wherein the method further comprises receiving financialinstrument price data indicating one or more market prices for thefinancial instrument, each of the one or more market prices beingpairwise contemporaneous or substantially contemporaneous withcorresponding ones of the option prices, and wherein the computing theset of two conditional extremal martingale probability distributions isfurther based at least in part on the financial instrument price data.19. The method of claim 11, wherein the method further comprisesreceiving a forward price or discount factor for the financialinstrument at the selected expiration date, and wherein the computingthe set of two conditional extremal martingale probability distributionsis further based at least in part on the forward price or the discountfactor.
 20. The method of claim 11, wherein the set of two conditionalextremal martingale probability distributions comprises: a firstmartingale probability distribution indicating lower bounds of possibleprices for an option on the financial instrument at the selectedexpiration date, the lower bounds being dependent on the option pricesfor the financial instrument; and a second martingale probabilitydistribution indicating upper bounds of possible prices for an option onthe financial instrument at the selected expiration date, the upperbounds being dependent on the option prices for the financialinstrument.
 21. One or more non-transitory computer-readable mediastoring computer-executable instructions which when executed by acomputer cause the computer to perform the method of claim
 11. 22. Amethod, comprising: with a computing device, receiving historical pricedata, the historical price data indicating one or more historical pricesfor a financial instrument; receiving maturity data, the maturity dataindicating one or more expiration dates, each of the one or moreexpiration dates being the expiration date for one or more options onthe financial instrument; selecting an expiration date from the one ormore expiration dates; receiving option prices, the option pricesindicating a market price for each of one or more available options onthe financial instrument, each of the option prices beingcontemporaneous or substantially contemporaneous with one another, andeach of the available options having an expiration date coinciding withthe selected expiration date; computing an approximating martingaleprobability distribution based at least in part on the historical pricedata, the selected expiration date, and the option prices, theapproximating martingale probability distribution indicatingprobabilities of possible prices for the financial instrument at theselected expiration date, the approximating martingale probabilitydistribution being indicative of option prices that approximate theoption prices, wherein the approximating martingale probabilitydistribution is determined by a fractional linear transformation of adistribution parameter function; and storing the approximatingmartingale probability distribution.
 23. The method of claim 22, furthercomprising computing estimated option prices from the approximatingmartingale probability distribution.
 24. The method of claim 22, whereinthe computing the approximating martingale probability distributioncomprises: computing initial estimates of the approximating martingaleprobability distribution based on test scalar values of the distributionparameter function; and computing the approximating martingaleprobability distribution based on test scalar values of an interpolationparameter function.
 25. The method of claim 22, wherein the computingthe approximating martingale probability distribution comprises:computing a representative martingale probability distribution using thehistorical price data, the representative martingale probabilitydistribution indicating probabilities of possible prices for thefinancial instrument at the selected expiration date; computing one ormore mathematical moments of the representative martingale probabilitydistribution; computing the fractional linear transformation of thedistribution parameter function, the fractional linear transformationbeing based at least in part on the one or more mathematical moments;assigning values to the distribution parameter function; assigningvalues to an interpolation parameter function; and computing theapproximating martingale probability distribution based at least in parton the fractional linear transformation, the values assigned to thedistribution parameter function, and the values assigned to theinterpolation parameter function.
 26. The method of claim 25, whereinthe one or more mathematical moments of the representative martingaleprobability distribution are computed using an Esscher transform of anempirical probability distribution.
 27. The method of claim 25, whereinthe approximating martingale probability distribution is based on theone or more mathematical moments of the representative martingaleprobability distribution, and not on any other use of the historicalprice data.
 28. The method of claim 22, wherein the computing theapproximating martingale probability distribution is performed usingonly algebraic manipulations.
 29. The method of claim 22, wherein thecomputing the approximating martingale probability distribution isperformed without using either stochastic computations or differentialequation computations.
 30. The method of claim 22, wherein the methodfurther comprises receiving financial instrument price data indicatingone or more market prices for the financial instrument, each of the oneor more market prices being pairwise contemporaneous or substantiallycontemporaneous with corresponding ones of the option prices, andwherein the computing the approximating martingale probabilitydistribution is further based at least in part on the financialinstrument price data.
 31. The method of claim 22, wherein the methodfurther comprises receiving a forward price or discount factor for thefinancial instrument at the selected expiration date, and wherein thecomputing the approximating martingale probability distribution isfurther based at least in part on the forward price or the discountfactor.
 32. One or more non-transitory computer-readable media storingcomputer-executable instructions, which when executed by a computercause the computer to perform the method of claim
 22. 33. A method,comprising: with a computing device, receiving historical price data,the historical price data indicating one or more historical prices for afinancial instrument; receiving a time horizon, the time horizon beingnot less than a shortest period between successive prices in thehistorical price data; computing one or more trading strategies based atleast in part on the historical price data and the time horizon, each ofthe one or more trading strategies comprising a function that relates avalue of the respective trading strategy to a price of the financialinstrument, wherein each of the one or more trading strategies isdetermined by a second order finite difference; and storing the one ormore trading strategies.
 34. The method of claim 33, further comprising,for each of the one or more trading strategies, computing an excessreturn, one or more hedging coefficients, or both an excess return andone or more hedging coefficients.
 35. The method of claim 33, whereinthe computing the one or more trading strategies comprises: computing atime horizon probability distribution indicating probabilities ofpossible prices for the financial instrument at the time horizon;computing one or more mathematical moments of the time horizonprobability distribution; and computing one or more coefficients of thesecond order finite difference, the one or more coefficients being basedat least in part on the one or more mathematical moments of the timehorizon probability distribution.
 36. The method of claim 35, whereinthe one or more mathematical moments are computed using an orthogonalseries method.
 37. The method of claim 33, wherein the time horizon is afirst time horizon, and wherein the computing the one or more tradingstrategies comprises: receiving a second time horizon, the second timehorizon being different than the first time horizon; receiving a forwardprice or discount factor for the financial instrument at the receivedsecond time horizon; computing a representative martingale probabilitydistribution using the forward price or the discount factor, therepresentative martingale probability distribution indicatingprobabilities of possible prices for the financial instrument at thesecond time horizon; computing one or more mathematical moments of therepresentative martingale probability distribution; and computing one ormore coefficients of the second order finite difference, the one or morecoefficients being based at least in part on the mathematical moments ofthe representative martingale probability distribution.
 38. The methodof claim 37, wherein the one or more mathematical moments of therepresentative martingale probability distribution are computed using anEsscher transform of an empirical probability distribution.
 39. Themethod of claim 33, wherein the one or more trading strategies arecomputed as orthogonal functions.
 40. The method of claim 33, furthercomprising computing hedging coefficients or excess returns associatedwith one or more of the trading strategies as explicit functions of aprice of the financial instrument.
 41. The method of claim 33, whereinthe trading strategies are computed without using option price data. 42.The method of claim 33, wherein the computing the one or more tradingstrategies is based on one or more mathematical moments of the timehorizon probability distribution, and not on any other use of thehistorical price data.
 43. The method of claim 33, wherein the computingthe one or more trading strategies is based on one or more mathematicalmoments of the representative martingale probability distribution, andnot on any other use of the historical price data.
 44. The method ofclaim 33, wherein the computing the one or more trading strategies isperformed using only algebraic manipulations.
 45. The method of claim33, wherein the computing the one or more trading strategies isperformed without using stochastic computations or differential equationcomputations.
 46. The method of claim 33, wherein the time horizonprobability distribution is based at least in part on the historicalprice data.
 47. One or more non-transitory computer-readable mediastoring computer-executable instructions, which when executed by acomputer cause the computer to perform the method of claim
 33. 48. Amethod, comprising: with a computing device, receiving historical pricedata, the historical price data indicating one or more historical pricesfor a financial instrument; receiving maturity data, the maturity dataindicating a maturity date for one or more options on the financialinstrument; receiving contemporaneous price data, the contemporaneousprice data indicating a contemporaneous or substantially contemporaneousmarket price for each of one or more available options on the financialinstrument, each of the one or more available options having a maturitydate coinciding with the received maturity date; computing a set of twoor more extremal martingale probability distributions, a set of two ormore conditional extremal martingale probability distributions, and oneor more approximating martingale probability distributions based atleast in part on the historical price data, the maturity data, and thecontemporaneous price data, wherein each of the two extremal martingaleprobability distributions, each of the two conditional extremalmartingale probability distributions, and each of the one or moreapproximating martingale probability distributions is determined by afractional linear transformation of a distribution parameter function;and causing the set of two or more extremal martingale probabilitydistributions, the set of two or more conditional extremal martingaleprobability distributions, and the one or more approximating martingaleprobability distributions to be displayed on a display device.
 49. Themethod of claim 48, further comprising: receiving a time horizon, thetime horizon being not less than a shortest period between successiveprices in the historical price data; computing one or more tradingstrategies based at least in part on the historical price data and thetime horizon, each of the one or more trading strategies comprising afunction that relates a value of the respective trading strategy to aprice of the financial instrument; and causing the one or more tradingstrategies to be displayed on the display device
 50. The method of claim49, further comprising: for at least one of the one or more tradingstrategies, computing an excess return, one or more hedgingcoefficients, or both an excess return and one or more hedgingcoefficients; and causing the excess return, the one or more hedgingcoefficients, or both the excess return and the one or more hedgingcoefficients to be displayed on the display device.
 51. The method ofclaim 48, wherein an identification of the financial instrument and thematurity date are received from a user via a user interface.
 52. Themethod of claim 48, wherein the method further comprises receiving aforward price or discount factor for the financial instrument at theselected expiration date, and wherein the computing the set of two ormore extremal martingale probability distributions, the set of two ormore conditional extremal martingale probability distributions, and theone or more approximating martingale probability distributions isfurther based at least in part on the forward price or the discountfactor.
 53. One or more non-transitory computer-readable media storingcomputer-executable instructions, which when executed by a computercause the computer to perform the method of claim 48.